Permutative automorphisms of the Cuntz algebras \({\cal O}_{n}\) are in bijection with the stable permutations of [n]k. They are also the elements of the restricted Weyl group of \({\rm Aut}({\cal O}_{n})\) . In this note, we characterize a class of stable involutions of [n]2. More precisely, we prove [2, Conjecture 12.2, p. 60], thus providing a new family (with 6 degrees of freedom) of automorphisms of the Cuntz algebras \({\cal O}_{n}\) for any n > 1.
