  In this paper we describe the fundamental group-scheme of a proper variety
fibered over an abelian variety with rationally connected fibers over an
algebraically closed field. We use old and recent results for the Nori
fundamental group-scheme, and of finite group-schemes in general to prove that
the kernel of such a fibration is finite, and that the homotopy exact sequence
holds in this case. As an application, we describe the fundamental
group-schemes of certain proper varieties that are connected by curves or are
related to them.
