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.
So my question, what groups have a simple automorphism group? And less preciously, what is going on when/if this happens?