Quantum automorphisms of twisted group algebras and free hypergeometric laws

We prove that we have an isomorphism of type $A_{aut}(\mathbb C_\sigma[G])\simeq A_{aut}(\mathbb C[G])^\sigma$,foranyfinitegroup$G$, and any 2-cocycle $\sigma$ on $G$. In the particular case $G={\mathbb{Z}}_n^2$, this leads to a Haar-measure preserving identification between the subalgebra of $A_o(n)$ generated by the variables $u_{ij}^2$, and the subalgebra of $A_s(n^2)$ generated by the variables $X_{ij}={\sum }_{a,b=1}^n{p}_{ia,jb}$. Since $u_{ij}$ is “free hyperspherical” and $X_{ij}$ is “free hypergeometric”, we obtain in this way a new free probability formula, which at $n=\mathrm{\infty }$ corresponds to the well-known relation between the semicircle law, and the free Poisson law.