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. 


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 always $q^{\alpha}$, it will be hard to get a factor of $q/2$ by cohomology calculations. But if we multiply by $2$, it's possible.