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We prove that we have an isomorphism of type $A_{aut}(\mathbbC_\sigma[G])\simeq A_{aut}(\mathbb C[G])^\sigma$, for any finite group $G$, andany 2-cocycle $\sigma$ on $G$. In the particular case $G=\mathbb Z_n^2$, thisleads 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 $X_{ij}=\sum_{a,b=1}^np_{ia,jb}$. Since$u_{ij}$ is "free hyperspherical" and $X_{ij}$ is "free hypergeometric", weobtain in this way a new free probability formula, which at $n=\infty$corresponds to the well-known relation between the semicircle law, and the freePoisson law.

Quantum automorphisms of twisted group algebras and free hypergeometric laws