I would like to ask the following: if for a group $G$ the homology $H_n(G,\mathbb{Z})$ is $\mathbb{Z}$-torsion for every $n\geq n_0$, then what can be said concerning $\mathbb{Z}$-torsion for $H_k(G,M)$ where M is a $\mathbb{Z}G$-module? For example I know that if M is a trivial $G$-module, then $H_n(G,M)\simeq H_n(G,\mathbb{Z})\otimes_{\mathbb{Z}}M\oplus Tor_1^{\mathbb{Z}}(H_{n-1}(G,\mathbb{Z}),M)$ [Weibel, Th. 6.1.12] and hence $H_n(G,M)$ is $\mathbb{Z}$-torsion if $H_n(G,\mathbb{Z})$ is. What happens if M is not a trivial $G$-module?

Weibel, "An introduction to homological Algebra"