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?


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