Let $M$ be a closed (compact and no boundary) topological manifold and $G=\mathbb{Z}/2\mathbb{Z}$ be the group of order two. Let $\mathcal{G}(M)$ be the space of continuous $G$-actions on $M$ (say, with the $C^0$ topology).
Question 1: What is $\pi_0(\mathcal{G}(M))$? In other words, how many path components does $\mathcal{G}(M)$ have?
Question 2: How does the answer compare in the smooth category (i.e. $M$ is a smooth manifold and all actions are smooth)?
If these questions are intractable in general, how about for special cases of $M$ like $M=S^n$ or $M=\mathbb{CP}^n$?
If $n=dim(M)= 2$ then the answers in topological and smooth categories are the same. The cardinality of $\pi_0({\mathcal G}(M)$ is finite for $M=S^2$ (two) and $M=RP^2$ (one) but is countably infinite otherwise. For $n\ge 3$ the answers in two categories are different. Already for $n=3$, I think, you have continuum of components of ${\mathcal G}_{top}(S^3)$ (because of wild involutions whose fixed-point sets are wild spheres and wild circles in $S^3$), while ${\mathcal G}_{diff}(S^3)$ has cardinality three, described by conjugacy classes of order 2 subgroups of $O(3)$.