  In recent work, Benkart, Klivans, and Reiner defined the critical group of a
faithful representation of a finite group $G$, which is analogous to the
critical group of a graph. In this paper we study maps between critical groups
induced by injective group homomorphisms and in particular the map induced by
restriction of the representation to a subgroup. We show that in the abelian
group case the critical groups are isomorphic to the critical groups of a
certain Cayley graph and that the restriction map corresponds to a graph
covering map. We also show that when $G$ is an element in a differential tower
of groups, critical groups of certain representations are closely related to
words of up-down maps in the associated differential poset. We use this to
generalize an explicit formula for the critical group of the permutation
representation of the symmetric group given by the second author, and to
enumerate the factors in such critical groups.
