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., strongly potentially characteristic subgroup)
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Statement

Verbal statement

A normal subgroup need not be strongly 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. Strongly potentially characteristic implies semi-strongly potentially characteristic
2. Normal not implies semi-strongly potentially characteristic

Proof

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