Finite groups of matrices over group rings

We investigate certain finite subgroups $\Gamma$ of $GL_{n}(\mathbf{Z}\Pi )$, where $\Pi$ is a finite nilpotent group. Such a group $\Gamma$ gives rise to a $\mathbf{Z}[\Gamma \times \Pi]$-module; we study the characters of these modules to limit the structure of $\Gamma$. We also exhibit some exotic subgroups $\Gamma$.