Continuous action by profinite completion of a group

Question

I'm working on a problem involving diagram dependence of a specific group action. It turns out that taking the profinite completion of the group will allow me to speak from a point of universality in some sense. Suppose $G$ is a group which acts continuously on a profinite space $X$. Is it true that the profinite completion $\hat{G}$ of $G$ also acts continuously on $X$?

Answer 1

No. For instance, let $G$ be any group that is not residually finite, let $X=\{0,1\}^G$ and let $G$ act on $X$ by permuting the coordinates by left translation. The action of $G$ on $X$ is faithful, but the canonical map from $G$ to its profinite completion is not injective, so the action cannot factor through the profinite completion. (More generally, this shows that any group can act faithfully on a profinite space, so having such an action doesn't tell you anything useful about the group on its own.)