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Note: I'm a physicist so this will be phrased somewhat in physics language.

Suppose we have a unitary modular tensor category $\mathcal{C}$. In physics language, we can think of $\mathcal{C}$ as representing a topological phase and the objects of the category as representing the quasiparticles of this phase.

In some cases, the fusion and braiding rules of $\mathcal{C}$ are invariant under the permutation of the quasiparticles (objects of $\mathcal{C}$) by the action of a finite group $G$. If the symmetry action is not anomalous then it is known that there exists a $G$-extension of $\mathcal{C}$ (see Fusion Categories and Homotopy Theory, 2009, Etingof et al)

My question: Given $\mathcal{C}$ and the action of $G$ on the objects of $\mathcal{C}$, does there exist a simple way of constructing the $G$-extension? In the physicist language, I would like a $G$-grading of the set of quasiparticles in the $G$-extension of $\mathcal{C}$.

Thanks!

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