Let $X$ be a smooth, projective curve defined over a finite field $k$. We denote by $\overline{X}$ it's extension to the algebraic closure $\overline{k}$. All the questions are about coherent sheaves.

Under what conditions can we say that a sheaf $\mathcal{F}$ on $\overline{X}$ comes, by tensorization, from a sheaf on $X$?

Is it true that if $\mathcal{F}$ on $\overline{X}$ is stable under the Frobenius morphism then it comes from a sheaf on $X$?

What about the extensions of two sheaves on $X$? Namely, let $\mathcal{F,G}$ be two sheaves on $X$. Is it true that the (sheaf)extensions of these two sheaves on $X$ are the extensions of $\mathcal{F}\otimes\overline{k}$ and $\mathcal{G}\otimes\overline{k}$ that are stable under the Frobenius?

What if, for two sheaves $\mathcal{F},\mathcal{G}$ on $X$ we have an extension $\mathcal{E}\otimes\overline{k}$ of $\mathcal{F}\otimes\overline{k}$ and $\mathcal{G}\otimes\overline{k}$, is it true that $\mathcal{E}$ is an extension of $\mathcal{F},\mathcal{G}$?

The two last questions look to me like some sort of descent for morphisms at the derived categories level.

Edit: I forgot to specify that "sheaf"="coherent sheaf" in this topic.

stableunder the Frobenius (explicit equations and an action on a graded module are fine with me). – Karl Schwede Jun 3 '10 at 7:46