Challenge: Can you perform any actual computations in group (co)homology?

[fingolfin]

For example, is there any way to use CAP_project or homalg to actually compute, say, the Schur multiplier $M(G) = H_2(G, Z)$ of a finite permutation group, or $H^2(G, Z)$ resp. $H^2(G, C^*)$?