\psi (z)^*\right\| < \frac{{\varepsilon}}{2}$ for all $x, y, z \in F$.

Using (\[Pr\_Magic\_a.2\]) at the first step, using (\[Eq1.2019.08.28\]) and (\[Pr\_Magic\_a.3\]) at the second step, and using (\[Pr\_Magic\_a.1\]) at the last step, we get $$\begin{aligned}
\label{Eq1.20200501}
\left\|a - \psi (v) b \psi (v)^* \right\|
&\leq
\left\| \psi \left(a\right) - \psi \left( v b v^*\right) \right\|
+
\left\| \psi \left( v b v^* \right) - \psi ( v ) \psi (b)  \psi ( v)^* \right\|
\\\notag
&<
  \frac{{\varepsilon}\| \psi \|}{2 {{\operatorname{card}}}(G)}
+
\frac{{\varepsilon}}{2} \leq {\varepsilon}.\end{aligned}$$ Therefore, using (\[Eq1.20200501\]) and Lemma \[PhiB.Lem\_18\_4\](\[PhiB.Lem\_18\_4\_10.a\]) at the first step, $$(a - {\varepsilon})_+ \precsim_{A} \psi ( v ) b  \psi ( v)^*  \precsim_A b.$$ This completes the proof.

\[Exten\_Ap\_in\] Let $A$ be a unital [C\*-algebra]{}, let $n \in {{\mathbb{Z}}_{> 0}}$ and let $\alpha \colon  G \to {{\operatorname{Aut}}}(A)$ be an action of a finite group $G$ on $A$ which is strictly approximately inner. Then it is easy to check that the action ${{\operatorname{id}}}_{M_n} \otimes \alpha \colon G \to {{\operatorname{Aut}}}(M_n (A))$ is also strictly approximately inner.

\[WC\_injectivity\] Let $A$ be a unital [C\*-algebra]{} and let $G$ be a finite group. Let $\alpha \colon G \to {{\operatorname{Aut}}}(A)$ be a strictly approximately inner action. Let $\iota \colon A \to {C^* 