Edit: I thought it might be helpful to point out the similarities to the Zariski topology: The Zariski topology basically encodes how prime ideals intersect. The fusion of a finite group encodes how Sylow subgroups intersect. Strong fusion not only keeps track of the intersections, but also of the (G-inner) maps between those intersections, so that the fusion becomes a category. Since fusion controls cohomology, it seemed natural to look at how fusion describes the classifying space of a group. Amazingly, it does a great job of describing the p-completion of the classifying space and facilitates fairly direct calculations. In other words, the data encoded by the "prime subgroups" (Sylow p-subgroups) also encodes a natural topological space associated to the group, its (p-completed) classifying space.
Several areas of combinatorics, like certain parts of graph theory and finite geometry, also seem to be based on the simple fact that interesting groups have interesting geometries. A recent classification of Steiner triple systems followed from detailed classifications of finite simple groups and multiply transitive permutation groups, and several families of graphs are interesting because of their automorphism groups.
I hope it is clear too that separating a group from its actions is not sensible. The actions of a group are encoded by the conjugacy classes of its subgroups, and it is entirely internal. Most geometries associated to groups are also internal. This is basically why the classification of finite simple groups can succeed: the natural action of a group is already contained inside it in an easy to describe way, so that once the local structure of a group is sufficiently similar to a known group, the group itself is isomorphic to a known group.