Strictly speaking, HQ Coy is distinct from Admin Coy in an inf unit. HQ Coy contains the command support bits and pieces e.g. sigs pl, the comd element itself, etc. It's where the command functions "live". Admin Coy is the sustainment piece of the unit e.g. supply, transport, maintenance, medical, etc. The term CSS (Combat Service Support) Coy is sometimes used interchangeably with Admin Coy.
In a Reserve unit, the terminology is generally the same, except that the functions of Admin and HQ Coys are often combined into a single organization simply because there usually aren't enough personnel to justify two separate sub-units (in some of the larger Reserve units, it may be different). In my unit, we operated a Rifle Coy (mission element), an Admin Coy (VERY basic CSS) and a HQ Section (essentially myself, the DCO, the RSM, the Adjt, and the Ops WO; our Orderly Room and Clerks were brigaded together under a Garrison OR, so we didn't "own" them).