Normal not implies strongly potentially characteristic

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) uses::Characteristic-potentially characteristic implies normal-potentially characteristic
 * 2) uses::Normal not implies normal-potentially characteristic

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