Fully invariant direct factor implies left-transitively homomorph-containing

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