\| \alpha_{ght^{-1}} (c^{*}_{t})  -  z_{ght^{-1}} c^{*}_{t} z^*_{ght^{-1}} \| \cdot \| z_{ght^{-1}} \|
\\
&\hspace*{3em} {\mbox{}}
<
 {{\operatorname{card}}}(G)^3 M^2 {\varepsilon}' + {{\operatorname{card}}}(G)^3 M^2 {\varepsilon}' = 2 {{\operatorname{card}}}(G)^3 M^2 {\varepsilon}' ={\varepsilon}.\end{aligned}$$ This completes the proof.

\[Pr\_Magic\_a\] Let $A$ be a unital [C\*-algebra]{} and let $\alpha \colon  G \to {{\operatorname{Aut}}}(A)$ be an action of a finite group $G$ on $A$ which is strictly approximately inner. Assume $a, b \in A_+$. Then $a \precsim_{A} b$ if and only if $a \precsim_{{C^* (G, A, {\alpha})}} b$.

We only need to prove the backwards implication. Let ${\varepsilon}> 0$. Let $a, b \in A_+$. By Lemma \[PhiB.Lem\_18\_4\](\[PhiB.Lem\_18\_4\_11\]), it suffices to show that $(a - {\varepsilon})_+ \precsim_{A} b$. We use $a  \precsim_{C^*(G, A, \alpha)} b$ to find $v \in C^*(G, A, \alpha)$ such that $$\label{Eq1.2019.08.28}
\left\| v b v^* - a \right\| < \tfrac{{\varepsilon}}{2 {{\operatorname{card}}}(G)}.$$ Set $F= \big\{b,  v, v^* \big\}$. Now, we apply [Proposition \[Prp\_magic\_a\]]{} with $\frac{{\varepsilon}}{2}$ and $F$ as given to get a unital surjective linear map $\psi \colon {C^* (G, A, {\alpha})}\to A $ such that:

1.  \[Pr\_Magic\_a.1\] $\| \psi \| \leq {{\operatorname{card}}}(G)$.

2.  \[Pr\_Magic\_a.2\] $\psi (c) = c$ for all $c \in A$.

3.  \[Pr\_Magic\_a.3\] $\left\| \psi (xy z^*) - \psi (x) \psi (y) 