  We develop the foundations of $G$-global homotopy theory as a synthesis of
classical equivariant homotopy theory on the one hand and global homotopy
theory in the sense of Schwede on the other hand. Using this framework, we are
able to construct the $G$-global algebraic $K$-theory of symmetric monoidal
categories with $G$-action, unifying equivariant algebraic $K$-theory, as
originally introduced by Shimakawa, and Schwede's global algebraic $K$-theory.
  As an application of the theory, we prove that the $G$-global algebraic
$K$-theory functor exhibits the symmetric monoidal categories with $G$-action
as a model of connective $G$-global stable homotopy theory, generalizing and
strengthening a classical theorem of Thomason. This in particular allows us to
deduce the corresponding statements for global and equivariant algebraic
$K$-theory.
