Stabilizers of continuous profinite action on a finite discrete set I need to prove the equivalence of categories of finite $G$-sets and the category of finite discrete set with a continuous action of the profinite completion of $G$. For any group G.
I already have a functor from the category of $G$-sets to the category $G^*$-sets(dont know how to place a hat on a letter in latex). In the following fashion:
Let $N_i$ be the normal subgroups of $G$ of finite index, then for any $G$-set S we have that $\prod_i S^{N_i}$ is a $G^*$-set, with the natural extension of the given action of G.
Now to reconstruct $S$ from $\prod_i S^{N_i}$(product of the fixed elements of $N_i$), I need that no points(or orbits) are lost. Since all orbits have a conjugacy class of stabilizers, I need that these classes consist of one subgroup(a normal one :)), so this product is a product of all orbits, not just a few.
How do I go about to prove this? Hints pls.
 A: I don't understand how $\widehat{G}=\varprojlim G/N$ is supposed to act on $\prod S^N$ (product of fixed point sets); in addition, the product $\prod S^N$ need not be finite, no? I can see that $\varprojlim G/N$ acts on $\varprojlim X/N$, but again we need a finite set. Instead let's reverse the direction of our desired natural isomorphism.
Given a $\widehat{G}$-set $X$ which is finite, discrete and has continuous action, since $G\subseteq\widehat{G}$ canonically we get a $G$-action on $X$ for free. This construction is functorial; if $X\to Y$ is $\widehat{G}$-equivariant then in particular it must be $G$-equivariant. Observe that if $X$ is a finite $G$-set then ${\rm Sym}(X)$ is finite and the kernel $N$ of $G\to{\rm Sym}(X)$ must be of finite index; given this information we can equip $X$ now with a $\widehat{G}$-action given by $\widehat{G}\to G/N\to{\rm Sym}(X)$. Check that this action is continuous if we impose the discrete topology on $X$, and that this construction is functorial, and is inverse to the other functor we have, thus giving us a pair of inverse functors ${\sf FinSet}_G\leftrightarrow {\sf DiscFinSet}_{\widehat{G}}$.
