Thurston geometries on bases of bundles of homogeneous spaces

Natural Seifert bundles for plesiocompact (in particular, compact) homogeneous spaces $M$ are considered. Conditions are given for the realizability of compact forms of Thurston geometries of dimension less than or equal to four on the bases $M_a^*$ of these bundles. The structure of these $M_a^*$ is discussed in detail.