How do I compute the group cohomology $H^2(G,A)$ if G is a finite abelian group acting nontrivially on a finite abelian group A?
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If $G$ is any group and $A$ is any $G$module, then $H^2(G,A)$ can be identified with the set of the equivalence classes of extensions $$1\to A\to H\to G\to 1$$ such that the action of $G$ on $A$ is the given action. Two extensions $H_1,H_2$ are said to be equivalent if there is an isomorphism $H_1\to H_2$ that makes the extension exact sequences commute. See K. Brown, Group cohomology, chapter 4. 


One can do the calculation using Kunneth theorem and the cohomology of cyclic group. See eqn J18 and appendix J.6 and J.7 in a physics paper http://arxiv.org/pdf/1106.4772v2 


You can compute it using the Bar resolution, see [Weibel, Hbook]. 

