You can have a group as a company limited by shares or by guarantee, or you can have it as an unincorporated body. I am treasurer of a group which is a Company. To be honest, there are no real advantages over a simple association and several disadvantages, such as having to file an annual return and accounts, all of which costs money, as well as having to generally comply witht the Companies Acts on matters such as meetings and resolutions.
With a company, the company itself usually owns the aircraft and the members of the group are shareholders if it is a company limited by shares or are simply 'members' of the company in a guarantee company. The company maintains its list of shareholders or members in a register. The company still has to notify the CAA of beneficial ownership, just as would an unincorporated group. In the latter, one or two people are registered with the CAA as trustees, though this is just for the purposes of the ANO and does not define the legal ownership. To qualify for cost sharing under the ANO, article 130, everyone has to have at least a 5% share, meaning no more than 20 members per group or company.
ChiSau1: I have just completely redran my own group's constitution so if you need something to work with let me know.