Characteristic not implies potentially fully invariant

Statement
It is possible to have a subgroup $$H$$ of a group $$G$$ such that $$H$$ is a characteristic subgroup of $$G$$ but is not a potentially fully invariant subgroup of $$G$$, i.e., there is no group $$K$$ containing $$G$$ such that  is a fact about::fully invariant subgroup of $$K$$.

Facts used

 * 1) uses::Normal not implies potentially fully invariant: There exists an example of a group $$J$$ and a normal subgroup $$H$$ of $$J$$ such that $$H$$ is not fully invariant in any group $$K$$ containing $$J$$.
 * 2) uses::NPC theorem: This states that if $$H$$ is a normal subgroup of $$J$$, there exists a group $$G$$ containing $$J$$ such that $$H$$ is a characteristic subgroup of $$G$$.

Proof
Let $$J$$ and $$H$$ be a group-subgroup pair as given by fact (1). By fact (2), there exists a group $$G$$ containing $$J$$ such that $$H$$ is a characteristic subgroup of $$G$$.

We want to show that there is no group $$K$$ containing $$G$$ such that $$H$$ is fully invariant in $$K$$. The reason for this is: if such a group $$K$$ exists, it would also contain $$J$$, so we'd have a group containing $$J$$ in which $$H$$ is fully invariant. This contradicts the choice of $$H$$ and $$J$$ as being examples for fact (1).