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. 


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


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 

