We investigate the \(\hbox {C}^*\) -algebra inclusions \(B \subset A \mathop {\rtimes _{\textrm{r}}}\Gamma \) arising from inclusions \(B \subset A\) of \(\Gamma \) - \(\hbox {C}^*\) -algebras. The main result shows that, when \(B \subset A\) is \(\hbox {C}^*\) -irreducible in the sense of Rørdam, and is centrally \(\Gamma \) -free in the sense of the author, then after tensoring with the Cuntz algebra \({\mathcal {O}}_2,\) all intermediate \(\hbox {C}^*\) -algebras \(B \subset C\subset A \mathop {\rtimes _{\textrm{r}}}\Gamma \) enjoy a natural crossed product splitting \(\begin{aligned} {\mathcal {O}}_2\otimes C=({\mathcal {O}}_2 \otimes D) \mathop {\rtimes _{{\textrm{r}}, \gamma , \mathfrak {w}}} \Lambda \end{aligned}\) for \(D:= C \cap A,\) some \(\Lambda <\Gamma ,\) and a subsystem \((\gamma , \mathfrak {w})\) of a unitary perturbed cocycle action \(\Lambda \curvearrowright {\mathcal {O}}_2\otimes A.\) As an application, we give a new Galois’s type theorem for the Bisch–Haagerup type inclusions \(\begin{aligned} A^K \subset A\mathop {\rtimes _{\textrm{r}}}\Gamma \end{aligned}\) for actions of compact-by-discrete groups \(K \rtimes \Gamma \) on simple \(\hbox {C}^*\) -algebras. Due to a K-theoretical obstruction, the operation \({\mathcal {O}}_2\otimes -\) is necessary to obtain the clean splitting. Also, in general 2-cocycles \(\mathfrak {w}\) appearing in the splitting cannot be removed even further tensoring with any unital (cocycle) action. We show them by examples, which further show that \({\mathcal {O}}_2\) is a minimal possible choice. We also establish a von Neumann algebra analogue, where \({\mathcal {O}}_2\) is replaced by the type I factor \(\mathbb {B}(\ell ^2(\mathbb {N})).\)
