In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group $G$ acting on $A$ without fixed point. I want to understand when there is a Hodge isometry $$\phi: H^{2*}(A/G,\mathbb{Z}) \to H^{2*}(A,\mathbb{Z}).$$

For any abelian surface $X$, the dimension of $H^k(X,\mathbb{Z})$ is ${4 \choose k}$, hence it is enough to show $\phi$ is injective.

Any comments/suggestions/references are very welcome!!!

In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group (may not be abelian group) $G$ acting on $A$ without fixed point. I want to understand when there is a Hodge isometry $$\phi: H^{2*}(A/G,\mathbb{Z}) \to H^{2*}(A,\mathbb{Z}),$$ where $\phi$ is the map induced from the natrual morphism $A \to A/G$.

For any abelian surface $X$, the dimension of $H^k(X,\mathbb{Z})$ is ${4 \choose k}$, hence it is enough to show $\phi$ is injective.

The background for this question is: according to Orlov, Mukai that $D(A),D(A/G)$ are derived equivalent iff there exists such Hodge isometry. I don't know if the polarization type is relevent or not, but that is what I have in my case.

Any comments/suggestions/references are very welcome!!!