For a group $G$, let Aut$(G)$ denote the group of all the automorphisms of $G$. Consider the following subgroups of Aut$(G)$:
Inn$(G)=$the group of inner automorphisms of $G$
IA$(G)=$ the group of those automorphisms of $G$ which induce identity map on $G/[G,G]$
Aut$_c(G)=$ the group of those automorphisms of $G$ which preserve conjugacy classes
Aut$_z(G)=$ the group of those automorphisms of $G$ which are identity on $G/Z(G)$.
These are normal subgroups of Aut$(G)$.
The natural question arises is: which of the above subgroups are characteristic?
I think that none of the above subgroups is always characteristic in Aut$(G)$. But for most of the well known (to graduates) groups, the above subgroups are characteristic. So the question comes, which I post here is,
Give example of a finite group (possibly of smallest order) in which one of the above subgroup is not characteristic in Aut$(G)$.