Every nontrivial characteristic subgroup is potentially characteristic-and-not-fully invariant

Statement
Suppose $$G$$ is a group and $$H$$ is a nontrivial characteristic subgroup of $$G$$. Then, there exists a group $$K$$ containing $$G$$ suchthat $$H$$ is a fact about::characteristic subgroup of $$K$$ but is not a fact about::fully invariant subgroup of $$K$$.

Related facts

 * Stronger than::Characteristic not implies fully invariant
 * Weaker than::Every nontrivial normal subgroup is potentially characteristic-and-not-fully invariant