 Solution

Can be false. Take the Klein four-group $G=C_2\times C_2$ and $H=C_2\times \{e\}$. It is a normal subgroup because $G$ is an abelian group. Both $H$ and $G/H$ have order two and are cyclic groups. But every nonidentity element of $G$ has order two, so none generates its four elements. Thus $G$ is not cyclic.