Do I understand that Inverness is a Class D Tower/CTR, with Class F airspace above it and then Class C, is this correct?
Well, no.
Inverness is a Class G Air Traffic Zone (tower controlled within 2.5nm of the Aerodrome Reference Point, up to 2000 feet AAL), in class G airspace, with class C overhead from FL195 to FL600. In that class G there are three levels of service from ATC - a Basic service, a Traffic service and a Deconfliction service.
There are proposals to increase that to class D within 25nm up to 5,500' AAL and E above that.