Central implies potentially characteristic

Statement
A central subgroup of a group is always a potentially characteristic subgroup: it can be realized as a characteristic subgroup inside a possibly bigger group.

Central subgroup
A subgroup $$H$$ of a group $$G$$ is termed a central subgroup if every element of $$H$$ commutes with every element of $$G$$, or equivalently, if $$H$$ is contained in the center of $$G$$.

Potentially characteristic subgroup
A subgroup $$H$$ of a group $$G$$ is termed a potentially characteristic subgroup if there exists a group $$K$$ containing $$G$$, such that, with the induced embedding, $$H$$ is a characteristic subgroup of $$K$$.

Similar facts

 * Finite NPC theorem, which states that any normal subgroup of a finite group is characteristic in some finite group containing it.
 * Finite normal implies potentially characteristic, which uses the fact that finite normal implies amalgam-characteristic.
 * Periodic normal implies potentially characteristic, which uses the fact that periodic normal implies amalgam-characteristic
 * Normal subgroup contained in hypercenter is potentially characteristic, which uses the fact that normal subgroup contained in hypercenter is amalgam-characteristic
 * Abelian implies every subgroup is potentially characteristic
 * Nilpotent implies every normal subgroup is potentially characteristic
 * Central implies potentially verbal in finite

Facts used

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

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