I tried to write a longer answer which froze, so I'll write a shorter version. You might look at the history of the "Solomon fusion system" which arose in a characterization problem undertaken by Ron Solomon in his work on the classification of finite simple groups.

I tried to write a longer answer which froze, so I'll write a shorter version. You might look at the history of the "Solomon fusion system" which arose in a characterization problem undertaken by Ron Solomon in his work on the classification of finite simple groups. This does not occur in a finite group, but was shown by Dave Benson to occur in a group like topological object ( "2-adic loop space") called BDI(4).

In some sense this led to work by topologists (especially Broto, Levi and Oliver) on "$p$-local finite groups" (actually topological spaces, not groups) which need to associate a linking system to a fusion system of a finite $p$-group. Aschbacher and Chermak showed in an Annals paper a few years ago that the Solomon fusion system does have an associated linking system, an therefore there is a $2$-local finite group associated to that fusion system. More recently, Chermak has shown that there is a $p$-local finite group associated to every saturated fusion system on a finite $p$-group.