airborne_artist is correct. Most non equity groups will in fact consist of an aircraft maintained to the standard of a public transport aircraft in which members buy a certain number of hours on terms set out in the rules of the group. This arrangement is not really a group in the more traditional sense, as you do not own a share of the aircraft, which you would have to do to legally cost share with a private category or permit aircraft.
Sometimes this route is used by owners who want to defray some of the costs of outright ownership. How legal they are in each case I wouldn't know but it seems to me the arrangement is essentially one of hire, so you cannot do it with a Permit aircraft or one which is maintained to a private category level.
The other way of doing it is to have a company with not more than 20 members. There each member holds at least 5%, which they can do even if they do not contribute 5% of the equity (value of the aircraft) when they join. This is a device to ensure that each member "owns" 5% because in reality the rules of the group do not generally allow mwmbers to sell their share for market value. Several groups I know of operate in this way, including my own Felthorpe group and the McAully Flying Group at Little Snoring.