Computing fusion rules for spherical G-extensions of fusion categories

Understand

A $G$-graded extension of a fusion category $\mathcal{C}$ yields a categorical action of $G$ on the center $Z(\mathcal C)$.

• If the extension admits a spherical structure, we provide a method for recovering its fusion rules in terms of the action.
• We then apply this to find closed formulas for the fusion rules of extensions of some group theoretical categories and of cyclic permutation crossed extensions of modular categories.