Let $G$ be a topological group and $\pi: P\rightarrow M$ a fiber bundle with fiber $G$ (working in continuous category). We say that $\pi$ is a principal G-bundle if
(1) $G$ acts freely on the right of $P$.
(2) For each $p\in M$, there exists a neighborhood $U$ of $p$ and a fiber preserving homeomorphism $\phi:\pi^{-1}(U)\rightarrow U\times G$ that is G-equivariant.
My question is, G-equivariantness of $\phi$ implicitly assumes that the action $G$ on $P$ restricts to an action $G$ on $\pi^{-1}(U)$. I was wondering if that is indeed always the case, or should there be a third alternative condition stating that to be the case?