Given a short exact sequence of finite abelian groups, is it possible to classify what groups can show up in the middle based on the kernel and the cokernel? I'm hoping the answer is much easier (than the group cohomology for the general case) in this case where everything is abelian, but I can't find a reference.
Or is it possible to formalize restricting the exact sequences to p-groups for a given prime p?
For example, if the kernel is $C_4^2$, the cokernel is $C_2^2\oplus C_6$, and the group in the middle has exactly 3 minimal generators, I can't think of examples where the group in the middle can be anything but $C_2\oplus C_8\oplus C_{24}$.