The term "inquiry" refers to a formal inquest (i.e., an investigation), whereas "enquiry" refers to "the act of questioning." However, there is still a lot of leeway in this difference. For example, you can use the word "inquiry" when talking about a survey conducted by someone who is not a police officer. In fact, according to the OED, the spelling of this word depends on its origin. If it comes from a legal source then we should use the spelling "inquiry"; but if it comes from a non-legal source then we can write "enquiry."" Non-legal examples of inquiry include: medical inquires, government inquiries, etc. Legal examples include: court inquiries, police inquiries. Thus, the correct spelling of this word is inquiry if it comes from a legal source and enquiry if it comes from a non-legal source.