This is a naïve question regarding etale cohomology.
I consider $\ell$-adic cohomology of algebraic varieties defined over an algebraic closure $\mathbb F_p$ with $p\not = \ell$. Assume given two projective varieties $X,Y$ together with a surjective proper map $$\pi:X\to Y.$$ Is there a systematic way to relate the cohomology groups $\mathrm H_c^{\bullet}(X,\mathbb Q_{\ell})$ and $\mathrm H_c^{\bullet}(Y,\mathbb Q_{\ell})$ together with the cohomology of the various fibers $X_y := \pi^{-1}(y)$ for $y\in Y$ ?
If moreover $X$ and $Y$, as well as the morphism $\pi$, are defined over $\mathbb F_q$ where $q$ is a power of $p$, is there a way to keep track of the Frobenius eigenvalues between these different cohomology groups ?