Fully invariant direct factor implies left-transitively homomorph-containing

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., fully invariant direct factor) must also satisfy the second subgroup property (i.e., left-transitively homomorph-containing subgroup)
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Statement

Suppose $H$ is a fully invariant direct factor of a group $G$. Then, $H$ is a left-transitively homomorph-containing subgroup of $G$: for any group $K$ in which $G$ is a homomorph-containing subgroup, $H$ is also a homomorph-containing subgroup.

Related facts

• Left-transitively homomorph-containing not implies subhomomorph-containing
• Right-transitively homomorph-containing not implies subhomomorph-containing
• Subhomomorph-containing implies right-transitively homomorph-containing