Rokhkin inclusions with integer and non-integer index

We construct new examples of inclusions of unital $C\sp*$-algebras of index-finite type with the Rokhlin property motivated by a broader attempt to understand the range of such inclusions beyond known models. In the course of this development, we observe an interesting phenomenon: inclusions with integer index, though not assumed to arise from group actions, exhibit internal behavior consistent with classical symmetry. In contrast, we construct inclusions with irrational index whose Rokhlin or tracial Rokhlin property arises from quantum symmetries - such as subfactor theory or more advanced tensor category action on Kirchberg algebras - and which cannot be modeled as fixed point algebras under any finite group action or finite dimensional Hopf $C\sp*$-algebra action. To our knowledge, these provide the first examples of inclusion with the tracial Rokhlin property not arising from a finite group action.