If a certain factor group is cyclic, can we transfer this to another factor group?

Question

Let $G$ be a finite non-abelian group with cyclic normal subgroup $N \lhd G$ such that $G/N$ is also cyclic. Let $H \lhd G$ be another cyclic normal subgroup of $G$ with $H \leq N$.

We know $G/N \cong (G/H)/(N/H)$.

Question: Can we infer that $G/H$ is also cyclic?

Answer 1

No. Take $G=S_3$, $N=\{e,(1\ 2\ 3),(1\ 3\ 2)\}$, and $H=\{e\}$. Then $G/N$ is cyclic, but $G/H\simeq G$, which is not cyclic.