My problem is the following, suppose you have a discrete group $G$ (finite type) and a $G$-module $M$, $M$ is a $\mathbb{Q}$-vectorial space. I would like to know if there are "ways" to compute $H^2(G,M)$, or just know if this is null. To make my question precise consider the following facts :
Because $G$ is a discrete group, I am talking about (discrete) group cohomology.The action of $G$ on $M$ can be anything.
The motivation of this question is clear, from the mathematical litterature, you know that $H^2(G,M)$ is meaningfull (it gives you all the extensions of $G$ by $M$, see for instance http://sierra.nmsu.edu/morandi/notes/GroupExtensions.pdf).
In general, Fox calculus has been proven useful to compute $H^1(G,M)$, see for instance part 3 of the following paper of Goldman : http://terpconnect.umd.edu/~wmg/SymplecticNature.pdf . Maybe some similar techniques can be applied to compute $H^2(G,M)$ (although I seriously doubt that a method as general as the Fox calculus for $H^2(G,M)$ can be found).
Of cours my main interest about this is to find cancellation (i.e. $H^2(G,M)=\{0\}$). I am looking for cases where this is true. Because I want you to understand the kind of answer I'd like, I will give you a gathering of "cancellation cases" you easily get studying this question :
When $G$ is a finite group then $H^2(G,M)=0$ (because it is a divisible module of $|G|$-torsion).
When $G=\mathbb{F}_k$ the free group over $k$ letters it is easily seen that $H^2(G,M)=0$ (because $cdg(\mathbb{F}_k)=1$ or because every extension of $\mathbb{F}_k$ can be splitted).
When $G$ is an extension of a finite group by a free group. It mixes both arguments above with a use of the Lyndon-Hochsild-Serre spectral sequence.
When $G$ is the fundamental group of a closed Riemann surface acting without fixed points on $M$. It is just about using Poincaré duality.
Of course, as I've tagged, just a reference about computing $H^2(G,M)$ for discrete group would do the job for me. May I add that I do know GAP and MAGMA, so please consider that I do not want to calculate $H^2(G,M)$ in some particular case (in which case I would indeed use my computer) but really trying to generate some more infinite series of examples...