Let $X$ be an (as nice as you prefer) alg. variety (or alg. stack) defined over $\mathbb{F}_{q}$ and let $\mathcal{F}$ be an l-adic sheaf on $X_n = X {\times_{\mathbb{F}q}} \mathbb{F}_{q^n}$. Fix an isomorphism between $\mathbb{C}$ and $\overline{\mathbb{Q}}_l$ (otw replace below $\mathbb{C}$ by the $l$-adic numbers). Suppose we know that the sheaf $\mathcal{F}$ comes (by pullback) from a sheaf, say $\mathcal{G}$, defined on $X$.

I would like to know if it is possible to recover the trace function of $\mathcal{G}$, say $g = f_\mathcal{G}:X(\mathbb{F}_q) \to \mathbb{C}$ defined by

$f_\mathcal{G}(x) = tr(Frob_x,\mathcal{G}_x)$ from the (analogously defined) trace function of $\mathcal{F}$, say

$f:=f_\mathcal{F}:X(\mathbb{F}_{q^n})\to\mathbb{C}$

I see that already for $X=\mathbb{F}_q$ you can only recover the trace function $g$ up to an $n$-th root of unity. So the following questions might look a bit silly:

- What's the best thing that we can do in general? i.e. if I have $f$ can I find $g$ up to some constant ($n$-th root of unity; need properness?).
- If I have $f$ what additional structure/information would I need to recover (uniquely) the function $g$?

Also, if you know any reference in which similar problems are treated please let me know.