Intermediate crossed product $C^{\ast }$-algebras

Let $B$ be a separable $C^{\ast }$-algebra, let $\mathrm{\Gamma }$ be a discrete countable group, let $\alpha :\mathrm{\Gamma }\to \text{Aut}(B)$ be an action, and let $A$ be an invariant subalgebra. We find certain freeness conditions which guarantee that any intermediate $C^{\ast }$-algebra $A{⋊}_{\alpha ,r}\mathrm{\Gamma }\subseteq C\subseteq B{⋊}_{\alpha ,r}\mathrm{\Gamma }$ is a crossed product of an intermediate invariant subalgebra $A\subseteq C_0\subseteq B$ by $\mathrm{\Gamma }$. Those are used to generalize related results by Suzuki.