Subhomomorph-containing implies right-transitively homomorph-containing

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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 with symbols

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

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