  For a braided fusion category $\mathcal{V}$, a $\mathcal{V}$-fusion category
is a fusion category $\mathcal{C}$ equipped with a braided monoidal functor
$\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given a fixed
$\mathcal{V}$-fusion category $(\mathcal{C}, \mathcal{F})$ and a fixed
$G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordinary fusion
category, we characterize the enrichments $\widetilde{\mathcal{F}}:\mathcal{V}
\to Z(\mathcal{D})$ of $\mathcal{D}$ which are compatible with the enrichment
of $\mathcal{C}$. We show that G-crossed extensions of a braided fusion
category $\mathcal{C}$ are G-extensions of the canonical enrichment of
$\mathcal{C}$ over itself. As an application, we parameterize the set of
$G$-crossed braidings on a fixed $G$-graded fusion category in terms of certain
subcategories of its center, extending Nikshych's classification of the
braidings on a fusion category.
