Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$, and suppose we have a surjective finite 'etale morphism $f:X\rightarrow Y$ (actually $Y=X/G$ for a free action of a finite group $G$), and an abelian sheaf $\mathcal{F}$ on the 'etale site $Y_{\'et}$ such that $f^*\mathcal{F}$ is a constant sheaf with stalk $\mathbb{Z}^r$ for some $r\geq 1$. What can be said about the sheaf $\mathcal{F}$? Can we conclude that $\mathcal{F}$ is also a constant sheaf?

I apologize beforehand if this is obvious... I am still in learning stage in etale cohomology. Any reference/suggestions regarding this would be appreciated.

Let $X$ and $Y$ be smooth projective varieties over $\mathbb{C}$, and suppose we have a surjective finite étale morphism $f:X\rightarrow Y$ (actually $Y=X/G$ for a free action of a finite group $G$), and an abelian sheaf $\mathcal{F}$ on the étale site $Y_{\mathrm{ét}}$ such that $f^*\mathcal{F}$ is a constant sheaf with stalk $\mathbb{Z}^r$ for some $r\geq 1$. What can be said about the sheaf $\mathcal{F}$? Can we conclude that $\mathcal{F}$ is also a constant sheaf?

I apologize beforehand if this is obvious... I am still in learning stage in étale cohomology. Any reference/suggestions regarding this would be appreciated.