In UK English 'inquiry' and 'enquiry' are often interchanged, but 'inquiry' is favoured when you are describing a formal investigation or a formal process of questioning. For instance, if the police were holding a formal investigation into a complaint, you will hear more often 'inquiry', and not 'enquiry'. "The police are holding an inquiry into the complaints made against the police officer" We mostly use 'enquiry' for a less formal questioning process. "I made an enquiry at the university to see if my application was received" ie. I asked some questions to see if they received my application. There is no sense of a formal investigation, I am simply asking some questions.

This noun ‘Enquiry’ can be used to denote a question or investigation about a subject matter. It is associated with the verb ‘to enquire’ which is the act of questioning. For example in a sentence,...’I would like to enquire about the condo for sale?’. The verb ‘enquire ‘ in this sentence is similar as ‘to pose a question for someone.’ Note that I added a question mark(?) on my example sentence. Likewise, ‘To inquire’ is basically to look into or examine with an investigation in detail. It could also mean, researching or survey for instance in a sentence....’The role of human rights commission is to inquire into the social injustices amongst the most vulnerable in the society.’