Let $A$ be a simple separable nuclear C$^*$-algebra with a unique tracial state and no unbounded traces, and let $\alpha$ be a strongly outer action of a finite group $G$ on $A$. In this paper, we show that $\alpha\otimes \mathrm{ id}$ on $A\otimes\mathcal{ W}$ has the Rohlin property, where $\mathcal{ W}$ is the Razak-Jacelon algebra. Combing this result with the recent classification results and our previous result, we see that such actions are unique up to conjugacy.
Categories: math.OA