Inn characteristic in Aut If $G$ is a centerless group then is $\mathrm{Inn}(G)$ necessarily characteristic in $\mathrm{Aut}(G)$? The condition of being centerless is necessary as $D_8$ provides a counterexample otherwise.

Answers from the comments (hence CW):

> This is only true if $\textrm{Aut}(G)$ is a complete group; that is, $\textrm{Aut}(G) \cong \textrm{Aut}(\textrm{Aut}(G))$ via the canonical homorphism $Aut(G)\to Aut(Aut(G))$. In particular, $G=S_{3} \times S_{3}$ is a counterexample. – Steve D

> @SteveD In fact it is true if and only if $\textrm{Aut}(G)$ is complete. – Derek Holt