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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$(G)$.

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    $\begingroup$ Isn't the dihedral group of order $8$ well known to graduate students? That's works for two of these cases. $\endgroup$ – Derek Holt Jun 5 '15 at 8:04
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    $\begingroup$ In fact it works for all four, because all four of them are equal to ${\rm Inn}(G)$ in that example. $\endgroup$ – Derek Holt Jun 5 '15 at 13:02

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