  For a finite group $G$, Turaev introduced the notion of a braided $G$-crossed
fusion category. The classification of braided $G$-crossed extensions of
braided fusion categories was studied by Etingof, Nikshych and Ostrik in terms
of certain group cohomological data. In this paper we will define the notion of
a $G$-crossed Frobenius $\star$-algebra and give a classification of (strict)
$G$-crossed extensions of a commutative Frobenius $\star$-algebra $R$ equipped
with a given action of $G$, in terms of the second group cohomology
$H^2(G,R^\times)$. Now suppose that $\mathcal{B}$ is a non-degenerate braided
fusion category equipped with a braided action of a finite group $G$. We will
see that the associated $G$-graded fusion ring is in fact a (strict)
$G$-crossed Frobenius $\star$-algebra. We will describe this $G$-crossed fusion
ring in terms of the classification of braided $G$-actions by Etingof,
Nikshych, Ostrik and derive a Verlinde formula to compute its fusion
coefficients.
