Examples of inclusions of unital $C\sp*$-algebras of index-finite type with the Rokhlin property

In this note, we provide several examples of inclusions of unital $C\sp*$-algebras of index finite-type with the Rokhlin property lacking in literature and more importantly we construct the example which does not belong to the fixed point algebra model which means that for the inclusion $P \subset A$ there is neither finite group action $ G \curvearrowright A$ such that $P=A^{G}$ nor finite dimensional $C\sp*$- Hopf algebra action $H \curvearrowright A$ such that $P=A^{H}$.
Comment: 12pages, 1 figure