Abelian group as second cohomology group of a pair G,M

Question

I'm currently studying group cohomology and in particular group extension; I'm trying to figure out a solution to the following problem: let A an abelian group, is possible to find a group G and a G-module M such that $H^2(G,M)\cong A$? If no, is it possible with the additional hypothesis that A is finitely generated? In other words, is safe to say that any (finitely generated) abelian group can occur as the second cohomology group of some pair (G,M) where G is a group and M is a G_module?

Answer 1

If $A$ is any abelian group, then $H^2(\mathbb{Z}\times\mathbb{Z},A)\cong A$, where $\mathbb{Z}\times\mathbb{Z}$ acts trivially on the coefficient group $A$. This follows from the universal coefficient theorem and the fact that $H_2(\mathbb{Z}\times\mathbb{Z},\mathbb{Z})\cong\mathbb{Z}$ and $H_1(\mathbb{Z}\times\mathbb{Z},\mathbb{Z})\cong\mathbb{Z}^2$.

(Similarly, for any $n\in\mathbb{N}$, $H^n(\mathbb{Z}^n,A)\cong A$.)