Strongly self-absorbing property for inclusions of $C^{\ast }$-algebras with a finite Watatani index

Let $P\subset A$ be a inclusion of unital C*-algebras and $E\text{\hspace{0.17em}⁣}:A\to P$ be a conditional expectation of index finite type. We introduce a Rokhlin property for $E$ and discuss about $\mathcal{D}$-absorbing proeprty, where $\mathcal{D}$ is a separable, unital, strongly self-absorbing C*-algebra. In this paper we consider permanent properties for strongly self-absorbing property under inclusions of unital C*-algebras with a finite Watatani index.