Potentially characteristic not implies normal-potentially characteristic

Statement
It is possible to have a potentially characteristic subgroup that is not a normal-potentially characteristic subgroup.

Weaker facts

 * Normal not implies normal-potentially characteristic
 * Normal not implies characteristic-potentially characteristic
 * Potentially characteristic not implies characteristic-potentially characteristic

Facts used

 * 1) uses::Potentially characteristic not implies normal-extensible automorphism-invariant, which in turn follows from uses::Normal not implies normal-extensible automorphism-invariant in finite and finite normal implies potentially characteristic.
 * 2) uses::Normal-potentially characteristic implies normal-extensible automorphism-invariant

Proof
The proof follows from facts (1) and (2).

Example of the dihedral group
Let $$G$$ be the dihedral group of order eight, and $$H$$ be one of the Klein four-subgroups.


 * $$H$$ is not a normal-potentially characteristic subgroup of $$G$$: Using the fact that every automorphism is center-fixing and inner automorphism group is maximal in automorphism group implies every automorphism is normal-extensible, every automorphism of $$G$$ can be extended to an automorphism of $$K$$ for any group $$K$$ containing $$G$$ as a normal subgroup. But since there is an automorphism of $$G$$ not sending $$H$$ to itself, $$H$$ cannot be characteristic in $$K$$.
 * $$H$$ is potentially characteristic in $$G$$: for instance, we can realize $$G$$ as the $$2$$-Sylow subgroup of the symmetric group of degree four, in such a way that $$H$$ becomes characteristic.