Inner automorphism group not is characteristic in automorphism group

Statement
The fact about::inner automorphism group of a group need not be a fact about::characteristic subgroup of the automorphism group.

Proof
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.