Automorphisms of the Bernoulli endomorphism and a class of skew-products

We consider a class of skew-products with base the one-sided two-shift equipped with the (1/2, 1/2) Bernoulli measure and fibre either Z(q) or the circle. We give conditions for the first kind to be isomorphic to the base itself and use this result to establish an isomorphism with the base for the second kind when a certain irrational is very well approximated by rationals. A consequence is that there is a circle action of automorphisms commuting with the one-sided two-shift.