Finite normal implies potentially characteristic

Statement
Suppose $$G$$ is a group and $$H$$ is a finite normal subgroup of $$G$$: $$H$$ is a normal subgroup of $$G$$ that is finite as a group. Then, there exists a group $$K$$ containing $$G$$ such that $$H$$ is characteristic in $$K$$.

Stronger facts

 * Finite NPC theorem: This states that a normal subgroup of a finite group can be realized as a characteristic subgroup in some finite group containing it.
 * NPC theorem: This states that any normal subgroup is potentially characteristic.

Facts used

 * 1) uses::Finite normal implies amalgam-characteristic
 * 2) uses::Amalgam-characteristic implies potentially characteristic

Proof
The proof follows directly by piecing together facts (1) and (2).