{\mathbb{B}}({\mathscr{H}}) \to A \otimes {\mathbb{B}}({\mathscr{H}})$ such that ${\left\| \varphi'(u)^*\varphi'(u)-1 \right\|}<\varepsilon$ and ${\left\| \varphi'(u)\varphi'(u)^*-1 \right\|}<\varepsilon$. Now $\tilde{u}:=\varphi' (u)|\varphi' (u)|^{-1/2}$ is a $G$-invariant odd unitary on ${\mathscr{H}}\otimes A$ which induces the $G$-equivariant isomorphism from ${\mathscr{H}}_0 \otimes A$ to ${\mathscr{H}}_1 \otimes A$.

Let $A$ be a ${\mathrm{C}^*}$-algebra and let $\{ \alpha _t\}$ be a homotopy of $G$-actions on $A$ such that $\alpha _0$ and $\alpha _1$ has finite (continuous) Rokhlin dimension with commuting tower. Then $(A,\alpha _0)$ and $(A, \alpha _1)$ are ${\mathrm{KK}}^G$-equivalent. Moreover, if $\alpha _0$ and $\alpha _1$ have Rokhlin property, the fixed point subalgebras $A^{\alpha_0}$ and $A^{\alpha _1}$ are ${\mathrm{KK}}$-equivalent.

It follows from Theorem \[thm:Rokinj\], Proposition \[prp:RokcRok\], Lemma \[lem:avbdl\] and Corollary 3.5 of [@mathOA150806815].

Continuous Rokhlin action of Hodgkin Lie groups
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In this section, we focus on the case that $G$ is a compact Lie group with Hodgkin condition, that is, $G$ is connected and $\pi _1(G)$ is torsion-free. An important feature of Hodgkin Lie groups is that the dual quantum group $\hat{G}$ is torsion-free in the sense of Section 7.2 of [@MR2563811]. Together with the strong Baum-Connes conjecture for ${\hat{G}}$ (Corollary 3.4 of [@MR2339371]), we obtain the following.

\[prp:BC\] Let $G$ be a Hodgkin Lie group. A ${\mathrm{KK}}^G$-morphism 