Inclusions of $C^*$-algebras arising from fixed-point algebras

We examine inclusions of $C^$-algebras of the form $A^H \subseteq A \rtimes\_{r} G$, 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 $A^H \subseteq A$ is $C^$-irreducible, in the sense that all intermediate $C^$-algebras are simple. We show that $A^H \subseteq A \rtimes\_{r} G$ 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.