Image-potentially direct factor equals central factor

Statement
The following are equivalent for a subgroup $$H$$ of a group $$G$$:


 * 1) $$H$$ is a central factor of $$G$$.
 * 2) There exists a group $$L$$ with a direct factor $$K$$ and a surjective homomorphism $$f:L \to G$$ such that $$f(K) = H$$.

Related facts

 * Direct factor implies central factor
 * Central factor not implies direct factor, central factor not implies complemented, complemented central factor not implies direct factor
 * Central factor satisfies image condition
 * Direct factor does not satisfy image condition
 * Direct factor implies right-quotient-transitively central factor
 * Direct factor implies join-transitively central factor
 * Join of finitely many direct factors implies central factor