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Suppose that we have an action of a group $G$ on a (quasi-)Hopf algebra $H$, so that we can construct $H\rtimes G$ as in Majid's Cross Products by Braided Groups and Bosonization. It is known that $Rep(H\rtimes G)\cong Rep(H)^G$ as tensor categories (see Galindo and Mombelli's paper, not sure if this was known before).

Now suppose additionally that $Rep(H)$ is a ribbon Ab-category with $G$-invariant R-matrix and twist element. Majid's results show that $Rep(H\rtimes G)$ is again a ribbon Ab-category. In particular both $Rep(H)$ and $Rep(H\rtimes G)$ can be "purified" as in Turaev's 1994 book (page 504).

Question: If I purify $Rep(H)$ and then $G$-equivariantize do I get the same thing (say as ribbon fusion categories) as purified $Rep(H\rtimes G)$?

It would be great to have a reference as I am sure I am not the first one to think about this.

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