Quotient-divisibility-faithful subgroup

Statement
A subgroup $$H$$ of a group $$G$$ is termed a quotient-divisibility-faithful subgroup if $$H$$ is a normal subgroup of $$G$$ and for any prime number $$p$$ such that the quotient group $$G/H$$ is $$p$$-divisible, $$G$$ is also $$p$$-divisible.

Facts

 * Derived subgroup is quotient-divisibility-faithful in nilpotent group