Perturbations of crossed product C*-algebras by amenable groups

We study uniform perturbations of crossed product C${}^{\ast }$-algebras by amenable groups. Given a unital inclusion of C${}^{\ast }$-algebras $C\subseteq D$ and sufficiently close separable intermediate C${}^{\ast }$-subalgebras $A$, $B$ for this inclusion with a conditional expectation from $D$ onto $B$, if $A=C⋊G$ with $G$ discrete amenable, then $A$ and $B$ are isomorphic. Furthermore, if $C\subseteq D$ is irreducible, then $A=B$.