P-localization of relative groups

A relative group is a pair (N,E) consisting of a group E and a normal subgroup N of E. Localization at sets of primes with respect to any radical is introduced as a functor on the category of relative groups. Given a radical R and a set of primes P, we consider the full subcategory of relative groups (N,E) which are P-local with respect to R, i.e. N is a P-local group and R(E/N) is trivial. We construct a left adjoint to the inclusion of this subcategory into the category of relative groups. As an application, when the radical R assigns the P′-isolator or the P′-torsion subgroup to a group, the effect of the classical P-localization is made explicit for groups fitting into an extension with a quotient F such that F/RF is torsion.