Extending localization functors (Q1903679)

An affirmative answer is given to the following question: Is there a terminal idempotent extension of \(P\)-localization of nilpotent groups (where \(P\) denotes a set of primes) over all groups? Using orthogonal pairs as main tool it is shown that, under suitable restrictions, an idempotent monad defined on a full subcategory of a category \({\mathcal C}\) can be extended to an idempotent monad on \({\mathcal C}\) in a universal way.