Given an algebraic surface $S$ with action of a finite group $G$. Is it true that the invariant lattice $H^2(X,\mathbb{Z})^G$ is generated by elements pulled back from the $H^2(X/G,\mathbb{Z})$ (or $H^2(\widetilde{X/G},\mathbb{Z})$)? This seems true for some examples such as an Enriques surface, but I don't know this works for any algebraic surface $S$.

I would appreciate it if someone could tell me what condition we need if this is not true in general.