There is an awful lot of misinformation being put about on this topic, isn't there!
For instruction towards the grant of a licence or a rating, you need a Public Transport C of A unless you are the sole owner of the aircraft (in which case Private Cat will do, but with some stipulations about maintenance). You need a licenced airfield.
Unless the instructor is not paid for the instruction. Paying an exhorbitant groundschool fee to him in lieu of airborne instruction time might be misunderstood by the CAA.
Once you have a licence, the various "renewal" checkrides can be in a group-owned private cat aircraft.
It's all in the AIC. I wrote to the CAA for clarification for our group, so we aren't in any doubt. We all do our renewal flights (and type signoff) in the group aircraft, and hire club ones for the issue of ratings.