for some $B$ holds under weaker assumption for the action. As in [@mathOA11072512], we can apply the Baum-Connes conjecture for the dual of Hodgkin Lie groups to the path of cocycle actions using the duality [@MR1970242] and we obtain any $G$-${\mathrm{C}^*}$-algebra is ${\mathrm{KK}}^G$-equivalent to $C(G) \otimes B$ for some $B$ if its dual action is asymptotically inner.

The following corollary is an analogue of Theorem 5.5 of [@mathOA14052469] for $\mathbb{T}$-${\mathrm{C}^*}$-dynamical systems.

\[cor:Krid\] Let $G$ be a Hodgkin Lie group and let $A$ be a $G$-${\mathrm{C}^*}$-algebra with continuous Rokhlin property. Then, there is a countable abelian group $M$ such that ${\mathrm{K}}$-groups ${\mathrm{K}}_*(A)$ is isomorphic to $M _*^n$ where $n:=2^{\operatorname{rank}G -1}$. Moreover, in this case $M \cong {\mathrm{K}}_0(A^\alpha) \oplus {\mathrm{K}}_1(A^\alpha)$.

It follows from Theorem \[thm:KKHod\]. We remark that $C(G)$ is ${\mathrm{KK}}$-equivalent to ${\mathbb C}^{n} \oplus C_0({\mathbb R})^n$, which follows from the fact that ${\mathrm{K}}^*(G)$ ($\ast =0,1$) are torsion-free abelian groups of rank $n$ (Theorem A (i) of [@MR0214099] and Hopf’s theorem, see for example Theorem 1.34 of [@MR2403898]) and the universal coefficient theorem [@MR894590].

\[cor:classify\] Let $G$ be a Hodgkin Lie group. Two $G$-${\mathrm{C}^*}$-algebras with continuous Rokhlin property are ${\mathrm{KK}}^{G}$-equivalent if and only if their fixed point algebras are ${\mathrm{KK}}$-equivalent. In particular, 