  Let $P \subset A$ be an inclusion of $\sigma$-unital C*-algebras with a
finite index in the sense of Izumi. Then we introduce the Rokhlin property for
a conditional expectation $E$ from $A$ onto $P$ and show that if $A$ is simple
and satisfies any of the property $(1) \sim (12)$ listed in the below, and $E$
has the Rokhlin property, then so does $P$.
  (1) Simplicity;(2) Nuclearity;(3) C*-algebras that absorb a given strongly
self-absorbing C*-algebra $\mathcal{D}$; (4)C*-algebras of stable rank one; (5)
C*-algebras of real rank zero;(6) C*-algebras of nuclear dimension at most $n$,
where $n \in Z^+$; (7)C*-algebras of decomposition rank at most $n$, where $n
\in Z^+$; (8) Separable simple C*-algebras that are stably isomorphic to AF
algebras; (9) Separable simple C*-algebras that are stably isomorphic to AI
algebras; (10) Separable simple C*-algebras that are stably isomorphic to AT
algebras; (11) Separable simple C*-algebras that are stably isomorphic to
sequential direct limits of one dimensional NCCW complexes; (12) Separable
C*-algebras with strict comparison of positive elements. In particular, when
$\alpha : G \rightarrow \rm{Aut}(A)$ is an action of a finite group $G$ on $A$
with the Rokhlin property in the sense of Nawata, the properties $(1) \sim
(12)$ are inherited to the fixed point algebra $A^\alpha$ and the crossed
product algebra $A \rtimes_\alpha G$ from $A$.
