  We 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.
