Inclusions of C'-algebras arising from fixed-point algebras

We examine inclusions of C* -algebras of the form A(H) subset of A (SIC)(r) G, where G and H are groups acting on a unital simple C* -algebra A by outer automorphisms and H is finite. It follows from a theorem of Izumi that A(H) subset of A is C *-irreducible, in the sense that all intermediate C *- algebras are simple. We show that A(H) subset of A (SIC)(r) G is C* -irreducible for all G and H as above if and only if G and H have trivial intersection in the outer automorphisms of A, and we give a Galois type classification of all intermediate C* -algebras in the case when H is abelian and the two actions of G and H on A commute. We illustrate these results with examples of outer group actions on the irrational rotation C* -algebras. We exhibit, among other examples, C* -irreducible inclusions of AF -algebras that have intermediate C* -algebras that are not AF -algebras; in fact, the irrational rotation C* -algebra appears as an intermediate C* -algebra.