  Let $\Gamma$ be a finite group acting on a Lie group $G$. We consider a class
of group extensions $1 \to G \to \hat{G} \to \Gamma \to 1$ defined by this
action and a $2$-cocycle of $\Gamma$ with values in the centre of $G$. We
establish and study a correspondence between $\hat{G}$-bundles on a manifold
and twisted $\Gamma$-equivariant bundles with structure group $G$ on a suitable
Galois $\Gamma$-covering of the manifold. We also describe this correspondence
in terms of non-abelian cohomology. Our results apply, in particular, to the
case of a compact or reductive complex Lie group $\hat{G}$, since such a group
is always isomorphic to an extension as above, where $G$ is the connected
component of the identity and $\Gamma$ is the group of connected components of
$\hat{G}$.
