N-Bundles for N an extension of a finite group by an abelian group

Let $W$ be a finite group and $T$ be an abelian group. Consider an extension $0 \ra T \ra N \ra W \ra 0$. For a smooth projective curve $X$, we give a precise description of the fiber of the quotient by $T$ map $q_T: \cM_X(N) \ra \cM_X(W)$ as a torsor over an abelian variety. We also prove a result on Mumford groups.