The title says it all. Given a (proper smooth) scheme $X$ over Spec$\mathbb{F}_q$, is it possible to calculate the zeta function of $X$ by from its étale covers?

Like for $\mathbb{P}^n$ you can cover it by $(n+1)$ copies of $\mathbb{A}^n$ which has as $k$-fold intersection of the form $\mathbb{G}_m^k \times \mathbb{A}^{n-k}$, and counting points are like doing the Inclusion-exclusion principle. Now I'm wondering if it can be done in the étale topology.

More concretely, given a scheme and a finite étale cover of it, how are their zeta functions related?

(BTW, I know there is an easier way to calculate the zeta function of $\mathbb{P}^n$ by using an affine stratification, the above example is just to illustrate my question.)