Cocentral implies right-quotient-transitively central factor

Statement
Suppose $$G$$ is a group and $$H$$ is a cocentral subgroup of $$G$$. Then, if $$K$$ is a subgroup of $$G$$ containing $$H$$ such that $$K/H$$ is a central factor of $$G/H$$, $$K$$ is also a central factor of $$G$$.

Related facts

 * Central implies join-transitively central factor
 * Direct factor implies right-quotient-transitively central factor

Facts used

 * 1) uses::Cocentrality is upward-closed
 * 2) uses::Cocentral implies central factor

Proof
(essentially follows from facts (1) and (2)).