Suppose $X$ is a smooth projective variety defined over an arbitrary algebraically closed field $k$, and consider the action of $\Sigma_n$ on the $n$-fold product $X^n$. Is it true that $H_{\acute{e}t}^i(\mathrm{Sym}^n(X),\mathbb{Q}_\ell)\cong H_{\acute{e}t}^i(X^n,\mathbb{Q}_\ell)^{\Sigma_n}$? In particular, what happens in the case where $\operatorname{char} k=p>0$?

In Grothendieck's Toh\^oku paper Sec. 5.2, he determines sufficient conditions to ensure that, for a topological space $X$ with a finite group $G$ acting on it (not necessarily faithfully), $H^i(X/G,\mathcal{A})\cong H^i(X,\mathcal{A})^G$ for a sheaf $\mathcal{A}$ (Cor. to Prop. 5.2.3). In characteristic zero, comparison theorems allow me to appeal to this result. In positive characteristic, if the variety lifts to characteristic zero, then I can make the same argument, but it seems like there ought to be a direct proof of this fact.

I am particularly interested in when $X$ is a surface, but would be happy to know of any general results (with references) similar to Grothendieck's result above.

stackquotient. When $\mathcal F = \mathbb Q_\ell$, (or more generally a ring where the order of $G$ is invertible) the cohomology of the quotient stack coincides with the cohomology of the scheme-theoretic quotient $X/G$ (the coarse moduli space). $\endgroup$ – Dan Petersen Sep 26 '18 at 21:24