Surjectivity need not be preserved by group localizations

Several examples studied in the literature motivate the question of whether or not all idempotent functors in the category of groups carry surjective homomorphisms into surjective homomorphisms. We negatively answer this question by giving a necessary and sufficient condition under which an idempotent functor on groups preserves surjectivity, and by displaying counterexamples which do not satisfy this condition. Our examples yield subcategories of groups, where epimorphisms need not be surjective.