P-localizing group extensions with a nilpotent action on the kernel

Assume P is a family of primes, and let ()_P represent the P-localization functor. If 1 → N → ^l G → ^∈ Q → 1 is a group extension giving rise to a nilpotent action of G on N, we prove that the sequence N_P → ^lP G_P → ^∈P Q_P → 1 is exact. Moreover, in the case where Q satisfies a certain pair of homological conditions, we show that the map ι_P is an injection. This generalizes the well-known result that ()_P is exact in the category of nilpotent groups. Applications are given to calculating P-localizations of virtually nilpotent groups.