Equivalence bundles over a finite group and strong Morita equivalence for unital inclusions of unital $C^{\ast }$-algebras

Let $\mathcal{A}=\{A_t{\}}_{t\in G}$ and $\mathcal{B}=\{B_t{\}}_{t\in G}$ be $C^{\ast }$-algebraic bundles over a finite group $G$. Let $C={\oplus }_{t\in G}{A}_t$ and $D={\oplus }_{t\in G}{B}_t$. Also, let $A=A_e$ and $B=B_e$, where $e$ is the unit element in $G$. We suppose that $C$ and $D$ are unital and $A$ and $B$ have the unit elements in $C$ and $D$, respectively. In this paper, we shall show that if there is an equivalence $\mathcal{A}-\mathcal{B}$-bundle over $G$ with some properties, then the unital inclusions of unital $C^{\ast }$-algebras $A \subset C$and$B \subset D$inducedby$$\mathcal{A}$$and$$\mathcal{B}$$ are strongly Morita equivalent. Also, we suppose that $\mathcal{A}$ and $\mathcal{B}$ are saturated and that $A^{\mathrm{\prime }}\cap C=\mathbf{C}1$. We shall show that if $A\subset C$ and $B\subset D$ are strongly Morita equivalent, then there are an automorphism $f$ of $G$ and an equivalence bundle $\mathcal{A}-{\mathcal{B}}^f$-bundle over $G$ with the some properties, where ${\mathcal{B}}^f$ is the $C^{\ast }$-algebraic bundle induced by $\mathcal{B}$ and $f$, which is defined by ${\mathcal{B}}^f=\{B_{f(t)}{\}}_{t\in G}$. Furthermore, we shall give an application.