Normal-potentially characteristic not implies characteristic

Statement
It is possible to have a group $$G$$ and a subgroup $$H$$ such that both the following hold:


 * $$H$$ is not a characteristic subgroup of $$G$$.
 * There exists a group $$K$$ containing $$G$$ as a normal subgroup and $$H$$ as a characteristic subgroup.

Facts used

 * 1) uses::subgroup of finite abelian group implies abelian-potentially characteristic

Proof
By fact (1), any subgroup of a finite abelian group can be realized as a characteristic subgroup in some bigger abelian group. Thus, if $$G$$ is a finite abelian group and $$H$$ is a non-characteristic subgroup of $$G$$, there exists some abelian group $$K$$ containing $$G$$ in which $$H$$ is characteristic. Note that $$G$$ is normal in $$K$$, because $$K$$ is abelian.

For instance, if we take $$G$$ to be the particular example::Klein four-group, and $$H$$ as one of the subgroups of order two, then we can take $$K$$ as a particular example::direct product of Z4 and Z2, containing $$G$$ as one of its cyclic subgroups.