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!!!