Question: Let $\varphi \colon G \to \text{Homeo}(X)$ be a group action on a topological space $X$ with basepoint $x_0$ and universal covering $\pi \colon \widetilde{X} \to X$. Then the subgroup of lifts $$ L = \{ (\widetilde{f},g) \in \text{Homeo}(\widetilde{X}) \times G : \widetilde{f} \circ \pi = \pi \circ \varphi(g)\} $$ constitutes a group extension $$ 1 \to \pi_1(X,x_0) \to L \to G \to 1, $$ where we map the homotopy class of a loop to the associated deck transformation in $L$.
Now a right split $\sigma \colon G \to L$ of this extension is equivalent to a lift of the group action $\varphi$ to $\widetilde X$.
If a lifted group action exists and furthermore the extension is trivial (i.e. a product), does the lifted group action commute with the action of the fundamental group by deck transformations?
Motivation/Background: If $\varphi(g)$ is homotopic to the identity for all $g$ and the fundamental group has trivial center, then one can construct a lifted group action using the homotopy lifting lemma and see that the lifted action commutes with the action of $\pi_1(X,x)$. Also in this case the extension is a product. So I wonder if this is just a special case for homotopy lifts or a general thing.
