In the literature concerning characteristic functions of varieties over a finite field, there is a notion called Weil structure defined in the following way:

Definition Let $X$ be a finite type scheme over $\overline{\mathbb{F}}_q$, and suppose $F$ is a geometric Frobenius endomorphism on $X$ over $\mathbb{F}_q$ (particularly, $X=X_0\times_{\mathrm{Spec}\ \mathbb{F}_q}\mathrm{Spec}\ \overline{\mathbb{F}}_q$ for some finite type scheme $X_0$ over $\mathbb{F}_q$). Denote by $\mathcal{L}$ a constructible $\overline{\mathbb{Q}}_{\ell}$-sheaf on $X$. Then an isomorphism $\phi\colon F^*\mathcal{L}\cong\mathcal{L}$ is called a Weil structure on $\mathcal{L}$. The Weil structure of an object in the derived category $D^b_c(X,\overline{\mathbb{Q}}_{\ell})$ is defined in the same manner.

My understanding is that, when $\mathcal{L}$ is the inverse image of a constructible $\overline{\mathbb{Q}}_{\ell}$-sheaf on $X_0$ along the natural projection $X\rightarrow X_0$, then it admits a natural Weil structure. My first question is whether the reverse is also true:

Question 1 Suppose $\mathcal{L}$ admits a Weil structure, does this imply $\mathcal{L}$ is the inverse image of a constructible $\overline{\mathbb{Q}}_{\ell}$-sheaf on $X_0$ along $X\rightarrow X_0$?

The other question concerning the pushforward with compact support:

Question 2 Suppose $\mathcal{L}$ admits a Weil structure. Let $f_0\colon X_0\rightarrow Y_0$ b a morphism of finite type schemes over $\mathbb{F}_q$, and denote by $f$ its base-change to $\overline{\mathbb{F}}_q$. Consider the derived functor $f_!$ from $D^b_c(X,\overline{\mathbb{Q}}_{\ell})$ to $D^b_c(Y,\overline{\mathbb{Q}}_{\ell})$, does the complex $f_!\mathcal{L}$ admits a canonical Weil structure?


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