Inclusions of unital $C^*$-algebras of index-finite type with depth 2 induced by saturated actions of finite dimensional $C^*$-Hopf algebras

Let $B$ be a unital $C^*$-algebra and $H$ a finite dimensional $C^*$-Hopf algebra with its dual $C^*$-Hopf algebra $H^0$. We suppose that there is a saturated action of $H$ on $B$ and we denote by $A$ its fixed point $C^*$-subalgebra of $B$. Let $E$ be the canonical conditional expectation from $B$ onto $A$. In the present paper, we shall give a necessary and sufficient condition that there are a weak action of $H^0$ on $A$ and a unitary cocycle $\sigma$ of $H^0 \otimes H^0 $ to $A$ satisfying that there is an isomorphism $\pi$ of $A\rtimes_{\sigma}H^0 $ onto $B$, which is the twisted crossed product of $A$ by the weak action of $H^0$ on $A$ and the unitary cocycle $\sigma$, such that $F=E\circ \pi$, where $F$ is the canonical conditional expectation from $A\rtimes_{\sigma}H^0 $ onto $A$.