Normal not implies strongly potentially characteristic

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., characteristic-potentially characteristic subgroup)
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Statement

Verbal statement

A normal subgroup need not be characteristic-potentially characteristic.

Statement with symbols

It is possible to have a group $K$ and a normal subgroup $H$ of $K$ such that there is no group $G$ containing $K$ in which both $H$ and $K$ are characteristic subgroups.

Facts used

1. Characteristic-potentially characteristic implies normal-potentially characteristic
2. Normal not implies normal-potentially characteristic

Proof

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