Does there exist a finite group whose automorphism group is simple? Ignore any trivial cases. Also, even if some examples exist for $|G|$ small, I still want to know if there are simple automorphism groups for arbitrarily large $|G|$. For such a group to exist, something catastrophically wrong must happen, as there are definitionally many, many normal subgroups of the automorphism group. See here for some examples. The inner automorphisms are the most famous. Universal power automorphisms are another. I've asked some recent questions about those. So my question, what groups have a simple automorphism group? And less preciously, what is going on when/if this happens?

Since $G/Z(G)$ is the inner automorphism group, and it is normal in $Aut(G)$, this has to be either trivial or all of $Aut(G)$. Thus $G$ is either abelian or a simple complete group. In the abelian case, you've already found a nice set of examples: $Z_2^n$ for $n\ge3$. For the simple complete groups, the Mathieu group $M_{23}$ works.