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.