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 $X$. 



What is known about lower etale cohomology of unirational varieties?
Concerning the $H^2$, for $X$ a smooth projective rationally chain connected variety over an algebraically closed field $k$ and $\ell \in k^\ast$, it follows from 
 Theorem 1.2 in <https://arxiv.org/abs/1703.05735> that



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