Let $M^3$ be a smooth three-manifold, and let $\gamma:G\to\operatorname{Homeo}(M)$ be a finite group action on $M$ by homeomorphisms.
Can $\gamma$ can be $C^0$-approximated by smooth group actions $\tilde\gamma:G\to\operatorname{Diff}(M)$?
Note that Bing and Moise proved (independently) that any homeomorphism $h:M\to M$ can be $C^0$-approximated by diffeomorphisms $\tilde h:M\to M$, however this does not a priori imply a positive answer to the question of approximation of group actions.
(I presume the answer to the analogous question in two dimensions is positive, but if this is not the case, that would be interesting as well.)
EDIT: Bing in this paper defined a continuous involution $\sigma:S^3\to S^3$ whose fixed set is a wildly embedded $S^2\hookrightarrow S^3$ (so, in particular, $\sigma$ is not topologically conjugate to a smooth involution). Bing also showed (in the very same paper!) that $\sigma$ is a $C^0$-limit of smooth involutions. Indeed, Bing considers a smooth involution $r:S^3\to S^3$ fixing a smooth $S^2\subseteq S^3$ and a small unknot $K\subseteq S^3$ stabilized by $r$ and intersecting the fixed locus transversally in two points. He then considers a sequence of diffeomorphisms $\varphi_n:S^3\to S^3$ which shrink the $n$th iterated Bing doubles $B^n(K)$ of $K$, namely every component of $\varphi_n(B^n(K))$ has diameter at most $\varepsilon_n>0$, where $\varepsilon_n\to 0$ as $n\to\infty$. Bing shows that (for judiciously chosen $\varphi_n$), the limit $\sigma:=\varphi_n\circ r\circ\varphi_n^{-1}$ exists and is the desired wild involution of $S^3$ (although, of course, the conjugating diffeomorphisms $\varphi_n$ do not converge to a homeomorphism, otherwise $\sigma$ would be topologically conjugate to $r$).
