A morphism between schemes is a *universal homeomorphism* if it is integral, surjective, universally injective. For morphism between algebraic stacks, this notion also make sense.

It is well know that a universal homeomorphism between schemes induces:

(1) homeomorphism between their underlying topological spaces; and

(2) equivalence between their etale sites.

I have seen the proof of these statements some time ago but didn't really understand well. So first a somewhat vague question:

**Question 1**: Is there a good *conceptual* reason why for schemes, homeomorphism between underlying topological spaces should imply equivalence between etale sites?

For me, at least psychologically, the underlying topological space and the etale site of a scheme seems quite unrelated. So I would prefer an intuitive explanation of this (seemingly?) coincidence.

Now for universal homeomorphism of stacks, one can also define its underlying topological space and its etale site and as far as I know, (1) is still true. It seems (2) may not be true, but something weaker is true, i.e. it induces equivalence of etale topoi, see for example Remark 4.26 of Behrend's book here.

In particular, if $\mathcal{X}$ is a (reasonable) Deligne-Mumford stack, the Keel Mori theorem says that it has a coarse moduli space $X$ and the natural morphism $\mathcal{X}\to X$ is a proper universal homeomorphism. (see the second paragraph in Conrad's paper).

Thus the etale topos of $\mathcal{X}$ is isomorphic to the etale topos of $X$. I'm trying to understand this isomorphism in the following special case:

Let a finite group $G$ acts on a ring $A$. Then my understanding is that the stack quotient $[(Spec A)/G]$ has a coarse moduli space $Spec(A^G)$, i.e. the GIT quotient. Then the etale topos of $[(Spec A)/G]$ is isomorphic to that of $Spec(A^G)$. In particular, when $A$ is an algebraically closed field with trivial $G$ action, then I reached a suspicious conclusion that the etale topos of $BG$ is isomorphic to etale topos of a point.

**Question 2:** Is this correct? If yes, what is the key (commutative algebra) ingredient behind this isomorphism of topos in this special case? Otherwise any comments on where I made mistakes are welcome.