I think it is more common to be a small investigaton, gathering info and asking questions. A police investigation would be an offical Police report, which might include many police work and time. Traditionally, enquire simply meant “ask,” while inquire was used for formal investigations. In the UK, the two words are used interchangeably, although inquire is still the more commonly used word for formal or official investigations. In the United States, inquire is the strongly preferred spelling in all uses.