Kind of hard to follow your train of thought, let me try to rephrase: You're asking whether the cohomology groups of K(G,1) with coefficients in M are torsion, where M is finite, and G is an infinite torsion group? Or are you asking something more general about Serre classes?
The first thing is trivially true, without any conditions on G. H^*(X;M) is always annihilated by the order of M.