Cocentral implies right-quotient-transitively central factor

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., cocentral subgroup) must also satisfy the second subgroup property (i.e., right-quotient-transitively central factor)
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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. Cocentrality is upward-closed
2. Cocentral implies central factor

Proof

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