the same argument, we prove that two Kirchberg $G$-algebras are $G$-equivariantly isomorphic if and only if they are ${\mathrm{KK}}^G$-equivalent. Together with Theorem \[thm:KKHod\], we obtain a complete classification of Kirchberg $G$-algebras (Theorem \[thm:Kir\]) in terms of equivariant ${\mathrm{KK}}$-theory, which is our second main theorem. An essential part in Theorem \[thm:Kir\] is construction of the model action ${\mathcal{O}}(G)$ of Kirchberg $G$-algebra with continuous Rokhlin property which is ${\mathrm{KK}}^G$-equivalent to $C(G)$. We construct it as a crossed product of the dual discrete quantum group $\hat{G}$ by an asymptotically representable action on ${\mathcal{O}}_\infty$. We use Kishimoto’s argument [@MR634163] in order to show that the crossed product is again a Kirchberg algebra.

Continuous Rokhlin property for compact group actions {#section:2}
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Throughout this paper, we assume that all topological groups are second countable Hausdorff and all ${\mathrm{C}^*}$-algebras are unital and separable unless otherwise noted.

Definition and Examples
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Let $G$ be a compact group and let $(A,\alpha)$ be a unital $G$-${\mathrm{C}^*}$-algebra. Set [$$\begin{aligned}
\mathfrak{T}_\alpha A&:=\{ a=(a_t) \in C_b([0,\infty),A) \mid \text{$(a_t)$ is uniformly $G$-continuous} \}, \\
\mathfrak{A}_\alpha A &:=\mathfrak{T}_\alpha A /C_0([0,\infty ) ,A), \ \mathfrak{C}_\alpha A:=\mathfrak{A}_\alpha A \cap A'.
 \end{aligned}$$]{} They are equipped with the canonical $G$-${\mathrm{C}^*}$-algebra structure. We say the (nonseparable) ${\mathrm{C}^*}$-algebra $\mathfrak{C}_\alpha A$ 