My question is essentially this: which parts of the representation theory of finite groups are really just applications of module theory, and which are not? Here is an example of each case. Induction of representations (at least for finite groups) is simply extension of scalars for modules over the group ring. Thus the definition of induction and its basic properties can be equally well studied for suitable class of modules with no mention of groups or representations. On the other hand, I have heard Mackey's irreducibility criterion listed as a classic example of a theorem that has no "good" module theoretic generalization because it involves double cosets, and it is very difficult to generalize double coset structure to modules.

Based on these examples I can break my question up into two, more specific questions: 1) What are some other examples of representation theoretic theorems that do or don't generalize well to a module theoretic setting? 2) When, in your opinion, is it useful for a representation theorist to work with the module theory, and when does this merely introduce unnecessary abstraction.