Twisted character varieties, covering spaces, and gerbes

For a finite-index subgroup $\widetilde\Pi$ of a finitely generated group $\Pi$, we establish various relations between (twisted) character varieties of $\Pi$ and $\widetilde\Pi$. We interpret our results geometrically in terms of the moduli space of flat connections on principal bundles twisted by a flat bundle gerbe over a manifold with the fundamental group $\Pi$.