Suppose $f: X\to Y$ is a finite map of varieties over a finite field $\mathbb F_q$. Is there an etale constructible $\mathbb Q_\ell$ sheaf $\mathscr F$ on $Y$ which counts the number of rational points of the form $f(x)$ for $x$ itself rational (as an application of the trace formula)?

If $f$ is closed, we can just use the pushforward. On the other hand,even if $X,Y$ are both spectra of fields, say of $\mathbb F_{q^n},\mathbb F_q$,then I am not sure what we want.

Edit 1: It seems to me that it might be easier to count $\deg(f)$ times the number of points and that's okay too. For instance, the number of squares in $\mathbb P^1$ is $(q-1)/2 + 2$ but since the eigenvalues are algebraic integers, we can't get a factor of $q/2$ by cohomology calculations. But if we multiply by $2$, it's possible.

Edit 2: In the case of $()^{\ell^n}: \mathbb G_m \to \mathbb G_m$, let us suppose we have a sheaf of the form we want and let the Frobenius have eigenvalues $q\alpha_i$ on degree zero compactly supported cohomology.

We can work out the number $\ell^n$ th powers in $\mathbb F_{q^r}$ and it depends on what power of $\ell$ divides $q^r-1$. If we further assume that $q-1$ is exactly divisible by $\ell$, we get equations of the form $\sum_i \alpha_i^r = \max(\ell^{n-k},\ell)$ for $\ell^{k-1}||r$.

We can solve this using Fourier inversion and it seems the unique solutions give us $\ell^{n-2k}$ copies of $\alpha_i = \zeta_{\ell^k}$ as $\zeta_{\ell^k}$ ranges over all primitive $\ell^k$ roots of unity and $1\leq k \leq n-1$ and $\ell^{n-1}$ copies of $1$.

Unfortunately $\ell^{n-2k}$ is not integral which seems to show that no $\mathbb Q_\ell$ sheaf that is defined by localizing a $\mathbb Z_\ell$ sheaf can work. Is there some other formalism of $\mathbb Q_\ell$ sheaves that can give us something useful?