Gauging the Categorical Connes' $\tilde{\chi}(M)$

We prove that if a finite group $G$ acts outerly on a McDuff $\rm II_1$ factor $M$, then $\mathsf{Rep}(G/KL)$ is a braided monoidal full subcategory of the categorical Connes' $\tilde{\chi}(M\rtimes G)$ defined in arXiv:2111.06378, where $K$ and $L$ are the centrally trivial and approximately inner parts in $G$ respectively. When $L$ is trivial, we give an explicit formula for the $G/K$-gauging procedure on $\tilde{\chi}(M\rtimes G)$. This is the categorical generalization of Connes' short exact sequence on $\chi(M\rtimes G)$. Using this machinery, for any finite group $G$, we construct a McDuff $\rm II_1$ factor $M$, whose $\tilde{\chi}(M)$ is braided equivalent to $\mathsf{Rep}(G)$. This is the first example of a braided fusion category which is not modular as $\tilde\chi$.