Preamble: Let $B$ be a manifold and $M \to B$ be a nontrivial fiber bundle on it with compact fiber. Let $\theta$ be a connection on $M \to B$. Let $H$ be a Lie group acting smoothly on $B$. If $H$ is simply connected, hopefully, I can take the fundamental vector fields of its action on $B$, lift them horizontally on $M$, and integrate them to get an action of $H$ on $M$ (I might be missing some hypotheses for the lifts of the vector fields to be integrable). However, when $H$ is not simply connected, which is my case of interest, there could be some holonomy and the action might not lift to $M$.
Question: I am looking for sufficient conditions for the action to lift.
Possible simplifications: I am interested in the case where $B$ and $M$ are complete simply-connected Riemannian manifolds for which the action is isometric, $M$ is a $G$-principal bundle and both $B$ and $M$ are simply connected. As a starting point, I am happy to assume $H$ and $G$ to be both $S^1$.