Rokhlin action on ${\mathcal{O}}_{|G|}$ is conjugate to the above action and hence has continuous Rokhlin property. In particular, the quasi-free action with respect to the left regular representation has continuous Rokhlin property (Proposition 5.6 of [@MR2053753]). Similarly, by the above example and Kirchberg’s absorption theorem (Theorem 3.2 of [@MR1780426]), the unique Rokhlin action of $G$ on $\mathcal{O}_2$ (Theorem 4.2 of [@MR2053753]) has continuous Rokhlin property.

Pick $\theta, \omega \in {\mathbb R}\setminus {\mathbb Q}$ such that $\theta-\omega \in {\mathbb R}\setminus {\mathbb Q}$ and let $$A:={\mathbb Z}{\, }_{\alpha _{\theta}} \!\! \ltimes(C(\mathbb{T}) \rtimes _{\alpha _\omega}{\mathbb Z})$$ be a noncommutative torus where we write $\alpha _z$ for the rotation automorphism on $C(\mathbb{T})$ (we use the same letter $\alpha _z$ for the automorphism $\alpha _t \rtimes {\mathbb Z}$ on $C(\mathbb{T}) \rtimes {\mathbb Z}$). Then, $A$ is simple by Theorem 1.9 of [@mathOA0609783] and the dual action $\hat{\alpha}_{\theta}$ of $\mathbb{T}$ has Rokhlin property since $\alpha _{\theta}$ is approximately representable (cf. Lemma \[lem:asymprep\]). Moreover, Proposition \[prp:injcRok\] implies that $(A \otimes {\mathcal{O}}_\infty, \alpha \otimes {\mathrm{id}})$ has continuous Rokhlin property (note that $A$ is ${\mathrm{KK}}^{\mathbb{T}}$-equivalent to $C(\mathbb{T}^3)$ and hence ${\mathfrak{J}}_{\mathbb{T}}$-injective).

Averaging technique via equivariant ${\mathrm{KK}}$-theory {#section:3}
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A fundamental technique for ${\mathrm{C}^*}$-dynamical systems with (continuous) Rokhlin property is an 