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: Looking at the example of $(.)^\ell: \mathbb G_m \to \mathbb G_m$, the number of $\mathbb F_{q^r}$ points depends on whether or not $\ell | q^r-1$ (it's either $(q^r-1)$ or $(q^r-1)\ell$ after multiplying by the degree). If we suppose that $q$ is a generator for $(\mathbb Z/\ell)^\times$, then it seems that on the degree zero compactly supported cohomology, we want the eigenvalues to be $q\zeta_{\ell-1}^{k}$ for $1 \leq k\leq \ell$ and similarly for degree $0$.