Given a complex manifold M equipped with an action of a group G, and a holomorphic principal
H–bundle EH on M, we introduce the notion of a connection on EH along the action of G, which is called a
G–connection. We show some relationship between the condition that EH admits a G–equivariant structure
and the condition that EH admits a (flat) G–connection. The cases of bundles on homogeneous spaces and
smooth toric varieties are discussed.