  We investigate the C*-algebra inclusions $B \subset A \rtimes_{\rm r} \Gamma$
arising from inclusions $B \subset A$ of $\Gamma$-C*-algebras. The main result
shows that, when $B \subset A$ is C*-irreducible in the sense of R{\o}rdam, and
is centrally $\Gamma$-free in the sense of the author, then after tensoring
with the Cuntz algebra $\mathcal{O}_2$, all intermediate C*-algebras $B \subset
C\subset A \rtimes_{\rm r} \Gamma$ enjoy a natural crossed product splitting
\[\mathcal{O}_2\otimes C=(\mathcal{O}_2 \otimes D) \rtimes_{{\rm r}, \gamma,
\mathfrak{w}} \Lambda\] 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 \[A^K \subset
A\rtimes_{\rm r} \Gamma\] for actions of compact-by-discrete groups $K \rtimes
\Gamma$ on simple 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 $\mathbb{B}(\ell^2(\mathbb{N}))$.
