Normal not implies normal-potentially relatively characteristic

Statement
A normal subgroup of a group need not be a normal-potentially relatively characteristic subgroup.

Facts used

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

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, $$H$$ is not invariant under this automorphism, which extends to an automorphism of $$K$$.
 * $$H$$ is normal in $$G$$: This is obvious.