  We introduce, for a symmetric fusion category $\mathcal{A}$ with Drinfeld
centre $\mathcal{Z}(\mathcal{A})$, the notion of
$\mathcal{Z}(\mathcal{A})$-crossed braided tensor category. These are
categories that are enriched over $\mathcal{Z}(\mathcal{A})$ equipped with a
symmetric tensor product, while being braided monoidal with respect to the
usual tensor product on $\mathcal{Z}(\mathcal{A})$.
  In the Tannakian case where $\mathcal{A}=\mathbf{Rep}(G)$ for a finite group
$G$, the 2-category of $\mathcal{Z}(\mathcal{A})$-crossed braided categories is
shown to be equivalent to the 2-category of $G$-crossed braided tensor
categories. A similar result is established for the super-Tannakian case where
$\mathcal{A}$ is the representation category of a finite super-group.
