  We study riemannian coverings $\varphi: \widetilde{M} \to \Gamma\backslash
\widetilde{M}$ where $\widetilde{M}$ is a normal homogeneous space $G/K_1$
fibered over another normal homogeneous space $M = G/K$ and $K$ is locally
isomorphic to a nontrivial product $K_1\times K_2$. The most familiar such
fibrations $\pi: \widetilde{M} \to M$ are the natural fibrations of Stieffel
manifolds $SO(n_1 + n_2)/SO(n_1)$ over Grassmann manifolds $SO(n_1 +
n_2)/[SO(n_1)\times SO(n_2)]$ and the twistor space bundles over quaternionic
symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The
most familiar of these coverings $\varphi: \widetilde{M} \to \Gamma\backslash
\widetilde{M}$ are the universal riemannian coverings of spherical space forms.
When $M = G/K$ is reasonably well understood, in particular when $G/K$ is a
riemannian symmetric space or when $K$ is a connected subgroup of maximal rank
in $G$, we show that the Homogeneity Conjecture holds for $\widetilde{M}$. In
other words we show that $\Gamma\backslash \widetilde{M}$ is homogeneous if and
only if every $\gamma \in \Gamma$ is an isometry of constant displacement. In
order to find all the isometries of constant displacement on $\widetilde{M}$ we
work out the full isometry group of $\widetilde{M}$, extending Elie Cartan's
determination of the full group of isometries of a riemannian symmetric space.
We also discuss some pseudo-riemannian extensions of our results.
