For any finite index subgroup $G$ there is a nonzero integer $m$ so that $G$ contains the elementary matrices $e_{ij}(m)$ that have ones on the diagonal, $m$ at the $(i,j)$ entry and zeroes elsewhere. So $\{gz-z: g\in G, z\in\mathbb Z^n\}$ contains all multiples of $m$
and ${\mathbb Z}^n_G$ is finite. Now just take a torsion-free finite index subgroup $G$.