$a \in A$ and all $g, h\in G$, $$\lim_{n \to \infty} \| \alpha_{g} (a) - u_n (g) a u_n (g)\|=0
\qquad
\mbox{ and }
\qquad
\lim_{n \to \infty} \| \alpha_{g} (u_n (h)) - u_n (g h g^{-1})\|=0.$$ Now take $z(g)= u_n (g)$ for some fixed and sufficiently large $n$. Therefore Condition (\[D\_9719\_AppRep.1’\]) through Condition (\[D\_9719\_AppRep.3’\]) in Definition \[D\_9719\_AppRep\_Phi\] hold.

Now, we omit Condition (\[D\_9719\_AppRep.3\]) in [Definition \[D\_9719\_AppRep\_Phi\]]{} and make the following definition.

\[D\_9719\_StrictAppInn\] Let $A$ be a unital [C\*-algebra]{}, let $G$ be a finite group, and let ${\alpha}\colon G \to {{\operatorname{Aut}}}(A)$ be an action of $G$ on $A$. We say that ${\alpha}$ is [*strictly approximately inner*]{} if for every finite set $F \subset A$ and every ${\varepsilon}> 0$, there are $z_g \in {{\operatorname{U}}}(A)$ for $g \in G$ such that:

1.  \[D\_9719\_StrictAppInn.1\] $\| {\alpha}_g (a) - z_g a z_g^* \| < {\varepsilon}$ for all $g \in G$ and all $a \in F$.

2.  \[D\_9719\_StrictAppInn.2\] $\| z_g z_h - z_{gh}\| < {\varepsilon}$ for all $g, h \in G$.

Obviously, strict approximate representability implies strict approximate innerness. But there is no reason to expect that the converse is true and the naive method to prove the converse fails. It seems that counterexamples are 