averaging process. Let $A$ be a separable $G$-${\mathrm{C}^*}$-algebra with continuous $X$-Rokhlin property. We use the same letter $\varphi$ for its $G$-equivariant completely positive contractive lift, which is a $G$-equivariant asymptotic morphism from $C(X)$ to $A$. Let us choose an increasing sequence $F_n$ of finite subsets of $A$ such that $\overline{\bigcup F_n}=A$ and families $\{ f_{n,i} \}_{i \in I_n}$ of positive continuous functions on $X$ as Lemma 4.2 of [@mathOA14075485]. Then, we obtain a $G$-equivariant completely positive asymptotic morphism $\xymatrix@C=1em{\psi :A \otimes C(X) \ar@{.>}[r] &A}$ defined by [$$\begin{aligned}
\psi _t(\xi ):=&(t-n) \sum _{i}\varphi _{\chi(t)}(f_{n,i})^{1/2}\xi(x_{n,i}) \varphi _{\chi (t)} (f_{n,i})^{1/2} \label{form:ab}\\
&+(n+1-t) \sum _i \varphi _{\chi (t)}(f_{n+1,i})^{1/2}\xi (x_{n+1,i}) \varphi _{\chi (t)} (f_{n+1,i})^{1/2} \notag  \end{aligned}$$]{} for $t \in [n,n+1]$ where $\chi : {\mathbb R}_{\geq 0} \to {\mathbb R}_{\geq 0}$ is a homeomorphism such that ${\left\| [\varphi_t(f_n) ,a]  \right\|} <2^{-n} |I_n|^{-1}$ for any $t \geq \chi (n)$ and $a \in F_n$. Note that $\psi ={\mathrm{id}}_A \times \varphi _\chi$ as a $\ast$-homomorphism from $A \otimes C(X)$ to $\mathfrak{A}_\alpha A$.

This averaging map is compatible with the picture of ${\mathrm{KK}}$-theory given in using completely positive asymptotic morphisms. Actually we have the isomorphism [$$\begin{aligned}
 {\mathrm{KK}}^G(A,B) \cong [\![ SA \otimes {\mathbb{K}}_G, SB \otimes {\mathbb{K}}_G ]\!] _{\mathrm{cp}}^G \label{form:KKcp} \end{aligned}$$]{} where 