  Let $G$ be an algebraic group and $\Gamma$ a finite subgroup of automorphisms
of $G$. Fix also a possibly ramified $\Gamma$-covering $\widetilde{X} \to X$.
In this setting one may define the notion of $(\Gamma,G)$-bundles over
$\widetilde{X}$ and, in this paper, we give a description of these objects in
terms of $\mathcal{H}$-bundles on $X$, for an appropriate group $\mathcal{H}$
over $X$ which depends on the local type of the $(\Gamma,G)$-bundles we intend
to parametrize. This extends, and along the way clarifies, an earlier work of
Balaji and Seshadri.
