The restricted Weyl group of the Cuntz algebra and shift endomorphism

It is shown that, modulo the automorphisms which fix the canonical diagonal MASA point-wise, the group of those automorphisms of O_n which globally preserve both the diagonal and the core UHF-subalgebra is isomorphic, via restriction, with the group of those homeomorphisms of the full one-sided n-shift space which eventually commute along with their inverses with the shift transformation. The image of this group in the outer automorphism group of O_n can be embedded into the quotient of the automorphism group of the full two-sided n-shift by its center, generated by the shift. If n is prime then this embedding is an isomorphism.