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 $f(x)$ (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.