Inclusion Criteria:  Patients must have histologically or cytologically confirmed recurrent or metastatic head and neck squamous cell carcinoma (HNSCC) that is considered incurable. Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as >= 20 mm (>= 2 cm) by chest x-ray or as >=10 mm (>= 1 cm) with computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam. Primary tumor locations of oral cavity, oropharynx, hypopharynx, and larynx are allowed. Participants may not have a primary tumor site of nasopharynx. Patients with oropharyngeal cancer must have known human papillomavirus (HPV) status defined by human papillomavirus type 16 (p16) testing. Patients should not have had prior systemic therapy administered in the recurrent or metastatic setting. Systemic therapy which was given as part of multimodal treatment for locally advanced disease is allowed. Patients must be able to provide an archival tissue specimen. Tumor tissue must have a documented combined positive score (CPS) of >= 1 for PD-L1. Age >= 18 years. Because no dosing or adverse event data are currently available on the use of ipatasertib in combination with pembrolizumab in patients < 18 years of age, children are excluded from this study. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 (Karnofsky >= 60%). Absolute neutrophil count >= 1,000/mcL. Platelets >= 100,000/mcL. Total bilirubin =< 1.5 x institutional upper limit of normal (ULN). Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT] =< 3 x institutional ULN. Creatinine =< 1.5 x institutional ULN. Glomerular filtration rate (GFR) >= 60 mL/min/1.73 m^2 (using the Cockcroft-Gault formula). Human immunodeficiency virus (HIV) infected patients on effective antiretroviral therapy with undetectable viral load within 6 months are eligible for this trial. For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral load must be undetectable on suppressive therapy, if indicated. Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load. Patients with treated brain metastases are eligible if follow-up brain imaging after central nervous system (CNS)-directed therapy shows no evidence of progression. Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association Functional Classification. To be eligible for this trial, patients should be class 2B or better. Patients must be able to swallow orally administered medication whole. For women of childbearing potential (WOCBP): agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive methods with a failure rate of < 1% per year during the treatment period, for 5 months after the last dose of pembrolizumab and 28 days after the last dose of ipatasertib, and agreement to refrain from donating eggs during this same period. For men: agreement to remain abstinent (refrain from heterosexual intercourse) or use contraceptive measures, agreement to refrain from donating sperm during the treatment period, and for 5 months after the last dose of pembrolizumab and 28 days after the last dose of ipatasertib. Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible.  Exclusion Criteria:  Prior treatment with a checkpoint inhibitor given for relapsed or metastatic disease. Prior treatment with a checkpoint inhibitor for locally advanced disease as part multidisciplinary treatment is allowed. History of malabsorption syndrome or other condition that would interfere with enteral absorption or result in the inability or unwillingness to swallow pills. Patient has a diagnosis of immunodeficiency or is receiving systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment. The use of physiologic doses of corticosteroids may be approved after consultation with the study principal investigator (PI). Active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids, or immunosuppressive drugs). Replacement therapy (e.g., thyroxine or physiologic corticosteroid replacement for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment.  Type 1 or Type 2 diabetes mellitus requiring insulin at study entry are ineligible.  Patients who are on a stable dose of oral diabetes medication >= 4 weeks prior to initiation of study treatment may be eligible for enrollment. Patients must meet the laboratory eligibility criteria for fasting blood glucose and hemoglobin A1c. Fasting glucose =< 150 mg/dL (8.3 mmol/L) and hemoglobin A1c =< 7.5% (58 mmol/mol). Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study. Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities > grade 1). Note: Patients with grade =< 2 neuropathy or grade =< 2 alopecia are an exception to this criterion and may qualify for the study. Note: If patients received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. Patients who are receiving any other investigational agents. Patients with known active CNS metastases and/or carcinomatous meningitis. Patients with previously treated brain metastases may participate provided they are radiologically stable, i.e., without evidence of progression for at least 4 weeks by repeat imaging (note that the repeat imaging should be performed during study screening), clinically stable and without requirement of steroid treatment for at least 14 days prior to first dose of study treatment. History of allergic reactions attributed to compounds of similar chemical or biologic composition to ipatasertib or hypersensitivity (grade >= 3) to pembrolizumab and/or any of the components of the solution for injection. Treatment with strong CYP3A inhibitors or strong CYP3A inducers within 2 weeks or 5 drug-elimination half-lives, whichever is longer, prior to initiation of study drug is prohibited. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product. Patients with uncontrolled intercurrent illness (including but not limited to interstitial lung disease or active, non-infectious pneumonitis) or a history of (non-infectious) pneumonitis that required steroids. Patients with psychiatric illness/social situations that would limit compliance with study requirements. Patients who are pregnant or breastfeeding, or are expecting to conceive or father children within the projected duration of the study, starting with the screening visit through 5 months after the last dose of study treatment. A WOCBP who has a positive urine pregnancy test (e.g., within 72 hours) prior to treatment will be excluded from the study. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. Pregnant women are excluded from this study because pembrolizumab is a monoclonal antibody agent and ipatasertib is an oral AKT inhibitor with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with pembrolizumab and ipatasertib, breastfeeding should be discontinued if the mother is treated with pembrolizumab or ipatasertib. Due to the potential risks, WOCBPs and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 5 months after the last dose of pembrolizumab and 28 days after the last dose of ipatasertib. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Patients with grade >= 2 uncontrolled or untreated hypercholesterolemia or hypertriglyceridemia are excluded. Patient has a known additional malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or carcinoma in situ (e.g., breast carcinoma, cervical cancer) that has undergone potentially curative therapy. History of or active inflammatory bowel disease (e.g., Crohn's disease and ulcerative colitis) or active bowel inflammation (e.g., diverticulitis). Lung disease: pneumonitis, interstitial lung disease, idiopathic pulmonary fibrosis, cystic fibrosis, aspergillosis, active tuberculosis, or history of opportunistic infections (pneumocystis pneumonia or cytomegalovirus pneumonia). Known clinically significant history of liver disease consistent with Child Pugh Class B or C, including active viral or other hepatitis (e.g., positive for hepatitis B surface antigen [HbsAg] or hepatitis C virus [HCV] antibody at screening), current drug or alcohol abuse, or cirrhosis.