Let $S$ be a smooth surface defined over a finite field $K$ of char. $p$. Let $G$ be a finite group of automorphisms of $S$. Let $Z\to S/G$ be the minimal resolution of the quotient of $S$ by $G$. Suppose that the fixed point set of every elements of $G$ is defined over $K$.

Let $\ell$ be a prime, $\ell\not=p$. Is it true that $$H^{i}(Z,\mathbb Q_\ell) \simeq H^{i}(S,\mathbb Q_\ell)^G$$ for $i=1,3$ and $$H^{2}(Z,\mathbb Q_\ell)\simeq H^{2}(S,\mathbb Q_\ell)^G+\mathbb Q_\ell C_1+\dots +\mathbb Q_\ell C_k$$where the $C_i$ are the exceptional curves of the resolution $Z\to S/G$ ?

This question is an echo of the question "Are there any known formulas about the Hodge-Deligne structure of quotients by action of groups ?" formulated in this forum 2 or 3 weeks ago.