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