OK – this is a tricky one, pretty much in the middle of change. However, for now, in British English …

Inquiry (noun) can be used like investigation.
The police have begun an inquiry/investigation into the suspicious death of an elderly person found on the riverbank last night.

An enquiry (noun) is a question, and to enquire (verb) is to ask for details about something.
“May I enquire about your health?” = “May I ask about your health?”

The jury is still out on this one, but I think we will see enquire disappear in favour of inquire, as is happening in the USA. Let’s see.

English: A Florentine policewoman handles a to...

English: A Florentine policewoman handles a tourist’s enquiry (Photo credit: Wikipedia)