  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.
