Pulling up the group action on the covering space

Question

Let $G$ a connected topological group (not necessarily simply-connected) such that $G$ acts transitively on the manifold $B$. let $\pi: E\to B$ be a covering space with finite fiber $\mathbb Z_p\not=\{0\}$. Can the action of $G$ be lift to $E$. I mean does $G$ acts transitively on $E$?

Answer 1

Consider the covering $\pi : S^3 \to \mathbb{RP}^3$. There is a transitive $SO(3)$-action on $\mathbb{RP}^3$ - in fact, $\mathbb{RP}^3$ is diffeomorphic to $SO(3)$, see here. On the other hand, $SO(3)$ cannot act transitively on $S^3$. To see this, note that if such an action were to exist, there would be a covering space $SO(3) \to S^3$, but this is impossible as $S^3$ is simply connected.