Non-connected Lie groups, twisted equivariant bundles and coverings

Let gamma be a finite group acting on a Lie group G. We consider a class of group extensions 1 -> G -> G -> gamma -> 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 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 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 G.