The metaphysics of groups

If you are a realist about groups (e.g. religious institutions, football teams, the Mafia etc.) there are three main theories of what to identify groups with. I offer reasons for thinking that two of those theories (groups as sui generis entities and groups as mereological fusions) fail to meet important desiderata. The third option is to identify groups with sets, which meets all of the desiderata if only we take care over which sets they are identified with. I then canvass some possible objections to that third theory, and explain how to avoid them.