Infinitesimal characterization of homogeneous bundles

Consider a principal bundle Q(B, H) on a base B which is compact and has finite fundamental group. We give necessary and sufficient conditions, in terms of the Atiyah sequence of Q(B, H), for Q(B, H) to be locally isomorphic to a bundle of the form G(G/S, S) for G a Lie group and S a closed subgroup of G. The proof involves the careful integration of certain infinitesimal actions of a Lie algebra on Q, B and the universal cover of B. By a homogeneous bundle we mean a principal bundle of the form G(G/H, H), where G is a Lie group and H is a closed subgroup. The infinitesimal properties of an arbitrary principal bundle Q(B, H, p) are encoded in its Atiyah sequence