Suppose *G*, *A*, and *B* are abelian groups with $i:A \to B$ an injective homomomorphism. Consider the groups $H^2(G,A)$ and $H^2(G,B)$ for the trivial action of *G* on *A* and *B*. *i* induces an injective homomorphism:

$$i_*: H^2(G,A) \to H^2(G,B)$$

The map $i_{\ast}$ is not always injective. For instance, setting *A* as the integers embedded in *B* the rationals, we see that the target group of $i_*$ is always trivial but the source group is often nontrivial (for instance, when *G* is finite cyclic).

Question: Under what conditions is $i_*$ injective and under what conditions is it an isomorphism? More specifically I am interested in the case where:

*G* is a finite abelian *p*-group and *A* and *B* are both finite cyclic *p*-groups with the order of *A* dividing the order of *B*. Here, *p* is a prime number.

Generic note: Note that the induced map at the level of cocycles (before going down to cohomology) is injective, but the trouble arises because it seems possible that some cocycles that were not coboundaries become coboundaries on enlarging the group of coefficients.