when these ${\mathrm{C}^*}$-algebras are in the UCT-class, then they are ${\mathrm{KK}}^{G}$-equivalent if and only if the ${\mathrm{K}}_*$-groups of fixed point algebras are isomorphic.

It follows from Lemma \[lem:avbdl\] and Theorem \[thm:KKHod\]. Note that $A$ is UCT if and only if so is $A^\alpha $ because $C(G)$ is ${\mathrm{KK}}$-equivalent to ${\mathbb C}^{n} \oplus C_0({\mathbb R})^n$.

Kirchberg $G$-algebras with continuous Rokhlin property {#section:4}
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In this section, we give a complete classification of Kirchberg $G$-algebras with continuous Rokhlin property up to conjugacy in terms of (equivariant) ${\mathrm{KK}}$-theory.

Preliminaries on duals of compact groups
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First, we collect some terminologies around the dual of compact groups, which are also used in Proposition \[prp:BC\] implicitly. We refer to [@MR3204665] for more information. Let $G$ be a compact group. Then, the group ${\mathrm{C}^*}$-algebra $C^*_\lambda(G)$ together with the nondegenerate $*$-homomorphism $${\hat \Delta}: C^*_\lambda(G) \to M(C^*_\lambda(G) \otimes C^*_\lambda(G))$$ determined by ${\hat \Delta}(\lambda_g) = \lambda _g \otimes \lambda _g$ for $g \in G$ is called the dual discrete quantum group ${\hat{G}}$ of $G$ (Definition 1.6.7 of [@MR3204665]). A *representation* of ${\hat{G}}$ (or a corepresentation of $G$) on a ${\mathrm{C}^*}$-algebra $A$ is a unitary $U \in M(C^*_\lambda(G) \otimes A)$ such that $$({\hat \Delta}\otimes {\mathrm{id}})(U) = U_{13} U_{23}.$$ The *regular 