Finite flat pushforward of a constant sheaf

Question

Let $A$ be an abelian group and consider the associated constant sheaf $A$ on a (smooth projective) variety $Y$ (over a field). Let $f: Y \to X$ be a surjective finite flat morphism. Is $f_*A$ also a constant sheaf on $X$?

Answer 1

Yes the image of a constant sheaf is a constant sheaf, even under significantly weaker hypotheses: it suffices to require that $Y$ be irreducible and $f$ surjective.

Actually the result is purely topological: given a surjective continuous map $f:Y\to X$ between topological maps, if $Y$ irreducible then the direct image of any constant sheaf $\underline A_Y$ on $Y$ is the constant sheaf $f_*(\underline A_Y)=A_X$ on $X$.
Indeed, given an arbitrary non-empty open subset $V\subset X$ we have $$\Gamma(V,f_*(A_Y))=\Gamma (f^{-1}(V),A_Y)=A$$ the last equality resulting from $f^{-1}(V)$ being connected, like any open subset of an irreducible space . This proves that $f_*( A_Y)=A_X$.

Warning: Beware that the result is false for non surjective $f$.
For example if $Y=*$ is a point, the direct image $f_*(A_Y)$ is the sky-scraper sheaf on $X$ with stalk $A$ at $f(*)\in X$