# Traces of Frobenius Endomorphism on Etale Cohomology and $G$-torsors

I have a smooth, projective, and rigid Calabi-Yau threefold $$X$$ defined over $$\mathbb{Q}$$. Such spaces always have integral models. Let's assume we have an action on $$X$$ by a finite abelian group $$G$$. The quotient is also a smooth, projective, and rigid Calabi-Yau threefold $$Y$$ which has an integral model. Therefore, interpreting $$G$$ as a group scheme, I believe we can think of $$X$$ as a $$G$$-torsor over $$Y$$:

$$0 \to G \to X \to Y \to 0.$$

So let's assume I know for all primes $$p>5$$ (say) the quantities

$$t_{3}(p)_{X} = \text{Tr}\big( \text{Frob}^{*}_{p}\big|_{H_{\text{et}}^{3}(\overline{X}, \mathbb{Q}_{\ell})}\big),$$

where $$H_{\text{et}}^{3}(\overline{X}, \mathbb{Q}_{\ell})$$ is the middle etale cohomology of the threefold $$X$$. By the modularity results of Noriko Yui and others, the $$t_{3}(p)_{X}$$ are the coefficients of a modular form.

My question is, do we have any technology allowing us to compute

$$t_{3}(p)_{Y} = \text{Tr}\big( \text{Frob}^{*}_{p}\big|_{H_{\text{et}}^{3}(\overline{Y}, \mathbb{Q}_{\ell})}\big),$$

for the quotient $$Y$$ given that it's related to $$X$$ through the above exact sequence? This might be way too naive, but I'm wondering if $$t_{3}(p)_{X}$$ and $$t_{3}(p)_{Y}$$ are related by group cohomology of $$G$$ or something.

If this is hopeless, then is there any technology for computing $$\#Y(\mathbb{F}_{p^{n}})$$ in terms of $$\#X(\mathbb{F}_{p^{n}})$$ given they're related by the above exact sequence?