$\mathbb{Z}$-torsion homology for groups 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
$$H_n(G,M)\simeq H_n(G,\mathbb{Z})\otimes_{\mathbb{Z}}M\oplus \operatorname{Tor}_1^{\mathbb{Z}}(H_{n-1}(G,\mathbb{Z}),M)$$ [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"
 A: There exists an acyclic group $G$ which has the property that there exists a finite-index normal subgroup $H\lhd G$ such that $\mathbb{Z} \leq  H_1(H;\mathbb{Z})$. In particular, then $H_1(G;\mathbb{Z}[G/H]) \cong H_1(H;\mathbb{Z})$ is not torsion (by Shapiro's lemma).
The example is the fundamental group of the complement of a wild arc. Examples in the linked paper, such as "Fox's stitch", are infinite cyclic covers of a hyperbolic 2-component link complement. There are such examples which are arithmetic, and therefore have a finite-index subgroup which is "RFRS", by a theorem of mine, in Criteria for virtual fibering. Thus, for any element in the group, there is a finite-index subgroup for which it is homologically non-trivial. This property passes to subgroups, in which case the fundamental group of the complement of Fox's stitch has the property that there is a finite-index subgroup with infinite abelianization (in fact, a non-trivial homomorphism to $\mathbb{Z}$).
A: This Kunneth formula still holds, I proved it here:
Kuenneth-formula for group cohomology with nontrivial action on the coefficient
which holds in our situation for all nontrivial $G$-modules.
