Subhomomorph-containment satisfies intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., subhomomorph-containing subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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Definition

Suppose $H\le K\le G$ are groups such that $H$ is a subhomomorph-containing subgroup of $G$. Then, $H$ is also a subhomomorph-containing subgroup of $K$.

Related facts

• Homomorph-containment satisfies intermediate subgroup condition