Group extensions whose relative p-localizations split

We investigate whether a group extension with nilpotent kernel and finitely generated quotient whose relative p-localization splits for every prime p must necessarily split itself. For the case where the kernel is abelian, three sets of conditions on the extension under which this holds are described. In order to demonstrate that these conditions are necessary, we present an example of a finitely generated solvable group extension with abelian kernel that splits at every relative p-localization but fails to split globally. The paper concludes with a discussion of extensions with non-abelian kernel.