Intermediate crossed product $C^*$-algebras

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