Subhomomorph-containing implies right-transitively homomorph-containing

This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Homomorph-containing subgroup (?) and Subhomomorph-containing subgroup (?)), to another known subgroup property (i.e., Homomorph-containing subgroup (?))
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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., subhomomorph-containing subgroup) must also satisfy the second subgroup property (i.e., right-transitively homomorph-containing subgroup)
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Statement

Statement with symbols

Suppose $H\leq K\leq G$ are groups. If $H$ is a homomorph-containing subgroup of $K$ and $K$ is a subhomomorph-containing subgroup of $G$.

Related facts

• Homomorph-containment is not transitive
• Subhomomorph-containment is transitive