Localizations associated to semidirect products

For homotopical localization with respect to any continuous map, there are results describing the relations among the localization functors associated to the maps of a given fibration. Here we treat an analogous question in a group-theoretical context: we study localization functors associated to a short exact sequence of groups. We further specialize to a split short exact sequence of groups. In particular, we describe explicitly the localization functors associated to a semidirect product of finitely generated Abelian groups.