Normal not implies normal-potentially characteristic

Verbal statement
It is possible to have a normal subgroup of a group that is not a normal-potentially characteristic subgroup.

Statement with symbols
We can have a group $$G$$ with a subgroup $$H$$ such that $$H$$ is normal in $$G$$, but whenever $$K$$ is a group containing $$G$$ as a normal subgroup, $$H$$ is not a characteristic subgroup in $$K$$.

Stronger facts

 * Weaker than::Normal not implies normal-potentially relatively characteristic

Weaker facts

 * Stronger than::Normal not implies characteristic-potentially characteristic

Facts used

 * 1) uses::Normal not implies normal-extensible automorphism-invariant
 * 2) uses::Normal-potentially characteristic implies normal-extensible automorphism-invariant

Proof
The proof follows directly 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 normal in $$G$$: This is obvious.