  The correspondence between the concept of conditional flatness and
admissibility in the sense of Galois appears in the context of localization
functors in any semi-abelian category admitting a fiberwise localization. It is
then natural to wonder what happens in the category of crossed modules where
fiberwise localization is not always available. In this article, we establish
an equivalence between conditional flatness and admissibility in the sense of
Galois (for the class of regular epimorphisms) for regular-epi localization
functors. We use this equivalence to prove that nullification functors are
admissible for the class of regular epimorphisms, even if the kernels of their
localization morphisms are not acyclic.
