Given a $\mathbb{C}$-scheme $S$, two $S$-schemes $X$ and $Y$ that are flat over $S$ and a coherent sheaf of $O_Y$-modules $F$.

Assume we have a (faithfully) flat $S$-morphism $\pi: X \rightarrow Y$ and we know that $\pi^{*}F$ is a coherent sheaf of $O_X$-modules that is flat over $S$.

Can we conclude that $F$ is flat over $S$?

Can we drop the faithfully? Or is this wrong in this generality, do we need to assume more, to get such a result?

My situation is the follwing $X=A\times S$, $Y=B\times S$, for two smooth projective schemes $A$ and $B$ and $F$ is of the form $\pi_{*}G$ for a coherent sheaf $G$ on $X$ flat over $S$, with the property $\pi^{*}\pi_{*}G\cong G$

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