Montoli, AndreaRodelo, DianaVan der Linden, Tim2017-04-072017-04-0720161370-1444AUT: DRO01261;http://hdl.handle.net/10400.1/9668We give a new sufficient condition for the normal extensions in an admissible Galois structure to be reflective. We then show that this condition is indeed fulfilled when X is the (protomodular) reflective subcategory of S-special objects of a Barr-exact S-protomodular category C, where S is the class of split epimorphic trivial extensions in C. Next to some concrete examples where the criterion may be applied, we also study the adjunction between a Barr-exact unital category and its abelian core, which we prove to be admissible.engjournal article