hard to find. However, there is evidence of existing such examples. Namely, assume the notation in the proof of Lemma \[Lem.PZ.20200623\]. Since $w_g^{(0)}$ is not a constant sequence, we cannot expect naively to get $\alpha_{\infty g} (w(h)) = w(g h g^{-1})$ from (\[D\_9719\_AppRep.1’\]) and (\[D\_9719\_AppRep.2’\]).

We provide a stronger version of Definition \[D\_9719\_StrictAppInn\] in the following lemma.

\[Lem.s.a.i\_hom\] Let $A$ be a [C\*-algebra]{}, let $G$ be a finite group, and let ${\alpha}\colon G \to {{\operatorname{Aut}}}(A)$ be an action of $G$ on $A$. Then ${\alpha}$ is strictly approximately inner if and only if for every finite set $F {\subset}A$ and every ${\varepsilon}> 0$, there is a [homomorphism]{} $g \mapsto z_g$ from $G$ to ${{\operatorname{U}}}(A)$ such that, for all $g \in G$ and all $a \in F$, $$\| {\alpha}_g (a) - z_g a z_g^* \| < {\varepsilon}.$$

It follows from semiprojectivity of $C^*(G)$.

The following lemma provides a version of Definition \[D\_9719\_StrictAppInn\] which is stated in terms of the sequence algebra.

Let $A$ be a separable unital C\*-algebra and let $G$ be a finite group. Let $\alpha \colon G \to {{\operatorname{Aut}}}(A)$ be an action. Then $\alpha$ is strictly approximately inner if and only if there exists a unitary representation $w$ of $G$ 