P-localizing group extensions with a finite kernel

Assume P is a family of primes, and let ()_P represent the P-localization functor. If 1 → N →^L G, → ^∈ Q → 1 is an exact sequence of groups with N finite, we prove that the sequence N_P^LP G_P →^∈P Q_P→ 1 is exact. Moreover, we provide an explicit description of Ker_iP when Q belongs to a specific class of groups defined by a cohomological property. This class contains all nilpotent groups, all free groups and all P-local groups, as well as certain extensions formed from these three types of groups. In conclusion, we discuss the implications of our results for the study of finite-by-nilpotent groups.