Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite characterstic $p\neq l$? Actually, I am interested in the case where $X$ is a quotient of a rational variety $Y$ by a free action of a finite group (whose order is prime to $p$). Does the situation differ much from the complex variety case; are there any examples where it does?

Certainly, my question is closely related to (the torsion) of $\operatorname{Pic}(X)$ and to the Brauer group of $X$.