Relative Gieseker's problem on $F$-divided bundles

Let $f: X\to Y$ be a proper surjective morphism of varieties defined over an algebraically closed field of positive characteristic. We prove that if $f$ has geometrically connected fibers then the induced homomorphism of $F$-divided fundamental groups is faithfully flat. An important new ingredient in our proof is an analogue of B. Bhatt's and P. Scholze's descent theorem \cite[Theorem 1.3]{Bhatt-Scholze2017} for $F$-divided bundles. As a corollary, we prove that in general if $X$ is normal, $Y$ is smooth, both $X$ and $Y$ are projective, and the induced map on \'etale fundamental groups is surjective, then the corresponding homomorphism on $F$-divided fundamental groups is faithfully flat. We also establish an analogous result for isomorphisms. This generalizes and strengthens a recent result of X. Sun and L. Zhang \cite{Sun-Zhang2025}, which in turn generalized earlier results of H. Esnault and V. Mehta \cite{Esnault-Mehta2010} and I. Biswas, M. Kumar, and A. J. Parameswaran \cite{Biswas-Parameswaran-Kumar2025}.