on $C$ and $D$ induced by $\widehat{\rho}$ and $\widehat{\sigma}$, respectively. Similarly, we define $\widehat{\widehat{\rho}}$ and $\widehat{\widehat{\sigma}}$, $C_2 =C_1\rtimes_{\widehat{\widehat{\rho}}}H$ and $D_2 =D_1 \rtimes_{\widehat{\widehat{\sigma}}}H$, respectively. Let $E_2^{\rho, u}$ and $E_2^{\sigma, v}$ be the canonical conditional expectations from $C_1$ and $D_1$ onto $C$ and $D$ defined in the same way as above. Also, let $E_3^{\rho, u}$ and $E_3^{\sigma, v}$ be the canonical conditional expectations from $C_2$ and $D_2$ onto $C_1$ and $D_1$ defined in the same way as above, respectively. Using $E_1^{\rho, u}$ and $E_1^{\sigma, v}$, we regard $C$ and $D$ as right Hilbert $A$-modules, respectively. Let ${\mathbf B}_A (C)$ and ${\mathbf B}_A (D)$ be the $C^*$-algebras of all right $A$-module maps on $C$ and $D$, respectively. We note that any right module map in ${\mathbf B}_A (C)$ or ${\mathbf B}_A (D)$ is adjointable by Kajiwara and Watatani [@KW1:bimodule Lemma 1.10] since $C$ and $D$ are of finite index in the sense of [@KW1:bimodule]. For any $x, y\in C$, let $\theta_{x, y}^C$ be the rank-one $A$-module map on $C$ defined by $\theta_{x, y}^C (z)=x\cdot {\langle}y, z {\rangle}_A$ for any $z\in C$. Similarly, we define $\theta_{x,y}^D $ the rank-one $A$-module map on $D$ for any $x, y\in D$. Then ${\mathbf B}_A (C)$ and ${\mathbf 