Inner automorphism group not is characteristic in automorphism group

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Statement

The Inner automorphism group (?) of a group need not be a Characteristic subgroup (?) of the automorphism group.

Proof

Further information: dihedral group:D8

Let G be the dihedral group of order eight. Then the automorphism group of G is also a dihedral group of order eight, with the inner automorphism group a Klein four-subgroup that is normal but not characteristic.