Right-quotient-transitively central factor implies join-transitively central factor

Verbal statement
Any right-quotient-transitively central factor is a join-transitively central factor.

Statement with symbols
Suppose $$H$$ is a normal subgroup of a group $$G$$ such that for any subgroup $$K$$ of $$G$$ containing $$H$$, such that $$K/H$$ is a central factor of $$G/H$$, $$K$$ is also a central factor of $$G$$.

Then, for any central factor $$N$$ of $$G$$, the join of subgroups $$\langle H,N \rangle$$, which in this case is also the product of subgroups $$HN$$ is also a central factor.

Converse

 * Join-transitively central factor not implies right-quotient-transitively central factor

Facts used

 * 1) uses::Central factor satisfies image condition

Proof
Given: $$H$$ is a normal subgroup of a group $$G$$ such that for any subgroup $$K$$ of $$G$$ containing $$H$$, such that $$K/H$$ is a central factor of $$G/H$$, $$K$$ is also a central factor of $$G$$.

To prove: For any central factor $$N$$ of $$G$$, the join of subgroups $$\langle H,N \rangle$$, which in this case is also the product of subgroups $$HN$$ is also a central factor.

Proof: Let $$K = HN$$, and consider the quotient map $$\rho:G \to G/H$$. Then, $$\rho(N) = K/H$$.


 * 1) $$K/H$$ is a central factor of $$G/H$$: By fact (1), since $$N$$ is a central factor of $$G$$, $$\rho(N)$$ is a central factor of $$\rho(G)$$, which translates to the above.
 * 2) $$K$$ is a central factor of $$G$$: This follows from the assumption about $$H$$ in the given data.

This completes the proof.