  We consider a homogeneous fibration $G/L \to G/K$, with symmetric fiber and
base, where $G$ is a compact connected semisimple Lie group and $L$ has maximal
rank in $G$. We suppose the base space $G/K$ is isotropy irreducible and the
fiber $K/L$ is simply connected. We investigate the existence of $G$-invariant
Einstein metrics on $G/L$ such that the natural projection onto $G/K$ is a
Riemannian submersion with totally geodesic fibers. These spaces are divided in
two types: the fiber $K/L$ is isotropy irreducible or is the product of two
irreducible symmetric spaces. We classify all the $G$-invariant Einstein
metrics with totally geodesic fibers for the first type. For the second type,
we classify all these metrics when $G$ is an exceptional Lie group. If $G$ is a
classical Lie group we classify all such metrics which are the orthogonal sum
of the normal metrics on the fiber and on the base or such that the restriction
to the fiber is also Einstein.
