Kunneth formula for group homology

Question

I'm trying to prove Kunneth formula for group homology. $$ 0 \to \bigoplus_p H_p(G,M)\otimes H_{n-p}(G',M') \to H_n(G\times G',M \times M') \to \bigoplus_p Tor_1^{\mathbb Z}(H_p(G,M),H_{n-p-1}(G',M')) \to 0 $$ I want to deduce it to Kunneth formula for chain complexes using the formula $(F\otimes_G M)\otimes (F' \otimes_{G'} M') \cong (F\otimes F') \otimes_{G\times G'}(M\otimes M'),$ ($F, F'$ are projective resolution of $\mathbb Z$ over $\mathbb ZG,\mathbb ZG'$ respectively) To use Kunneth formula for chain complexes I need $F\otimes_G M$ or $F' \otimes_{G'} M'$ to be complex of $\mathbb Z$-free modules. It's true if $G$ acts trivially on $M$ and $M$ is a free $\mathbb Z$- module, because I may choose F to be bar-resolution then $F_G$ is free complex. And $F\otimes_G M \cong F_G \otimes M$ is free module as a tensor product of free modules. But this seems to be too strong conditions, whether is it possible to do fewer assumptions?