Crossed $S$-matrices and Fourier matrices for Coxeter groups with automorphism

We study crossed $S$-matrices for braided $G$-crossed categories and reduce their computation to a submatrix of the de-equivariantization. We study the more general case of a category containing the symmetric category $\mathrm{Rep}(A,z)$ with $A$ a finite cyclic group and $z\in A$ such that $z^2=1$. We give two example of such categories, which enable us to recover the Fourier matrix associated with the big family of unipotent characters of the dihedral groups with automorphism as well as the Fourier matrix of the big family of unipotent characters of the Ree group of type ${}^2F_4$.