My question came up while reading this article by Nicholas Katz, specifically lemma 4.2. I don't think it's necessary to read the article to answer the question, but I'm including it anyways for reference.
Suppose we have a smooth, affine, geometrically integral curve $U_0/\mathbb{F}_{q}$, and let $U/\overline{\mathbb{F}}_{q}$ be it's base change. Then we have the following exact sequence:
$$1 \rightarrow \pi_{1}(U) \rightarrow \pi_{1}(U_{0}) \rightarrow Gal(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q}) \rightarrow 1$$
One has the frobenius $Frob_{q}\in Gal(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q})$, which acts on $H^{1}(U,\overline{\mathbb{Q}}_{\ell})$ by transport of structure.
In this proof, we know that $Frob_{q}$ does not have $1$ as an eigenvalue, so that it operates without fixed points. To prove his result he derives a contradiction by constructing a nontrivial continuous additive homomorphism $f:\pi(U_{0})\rightarrow \overline{\mathbb{Q}}_{\ell}$, and claims that by restriction to $\pi_{1}(U)$ we get a fixed point in $H^{1}(U,\overline{\mathbb{Q}}_{\ell})$.
I'm aware of the isomorphism $Hom_{cont}(\pi_{1}(U),\overline{\mathbb{Q}}_{\ell})\simeq H^{1}(U,\overline{\mathbb{Q}}_{\ell})$, and as far as I can tell the proof of the claim would go like this. Use the homomorphism $Gal(\overline{\mathbb{F}}_{q}/\mathbb{F}_{q}) \hookrightarrow Aut(\pi_{1}(U))/Inn(\pi_{1}(U))$ to get an action of $Frob_{q}$ on $\pi_{1}(U)$. For $\phi\in Hom_{cont}(\pi_{1}(U),\overline{\mathbb{Q}}_{\ell})$ define $(Frob_{q}\cdot\phi)(a) = \phi(a^{Frob_{q}})$. Because our $f$ mentioned above is restricted from $\pi_{1}(U_{0})$ where it also factors through $\pi_{1}(U_{0})^{ab}$, $f$ is indeed invariant under this action.
Essentially what I don't see is why the isomorphism $Hom_{cont}(\pi_{1}(U),\overline{\mathbb{Q}}_{\ell})\simeq H^{1}(U,\overline{\mathbb{Q}}_{\ell})$ commutes with the action of $Frob_{q}$, which seems to be necessary. I only know the above isomorphism through a chain of fairly non-constructive isomorphisms-- something along the lines of what's mentioned in 11.3 of Milne's notes. However, it seems unlikely that something like this wouldn't turn out to be true. Can anyone explain why?
