Let's assume $\mathcal{F}$ is a sheaf of abelian groups on the big etale site of $k$-schemes. As an example, we know that values of $\mathcal{F}$ on any Zariski open subset of $\mathbb{A}_{k'}^n$ (for some $n$ and any finite extension $k'$ of $k$) is trivial. What can be said about the values of $\mathcal{F}$ on a finite etale covers of $\mathbb{A}^n_k$ (let's say only degree $p^n$ covers)? Is it necessarily trivial?
You can assume that the char is $p$. This probably is not true with this generality so let's add some conditions like being a constructible sheaf. What if we replace a sheaf with a complex in the derived category of abelian sheaves and assume that the complex is quasi-isomorphic to some complex of constructible sheaves?