Let $G$ be a group having a core-free subgroup $H$ such that the interval $[H,G]$ in the subgroup lattice $\mathcal{L}(G)$ is [ACC][1] of infinite length, and for every $K \in (H,G]$, $G$ is generated by a single $K$-coset (i.e. there is $g \in G$ with $\langle Kg \rangle = G$).  

**Question**: Is $G$ generated by a single $H$-coset?

It is an exercise to reformulate as:  let $I$ be an infinite set, $[H,G]$ ACC interval of groups and $(K_i)_{i \in I}$ its [coatoms][2], i.e. the maximal elements in $[H,G)$. Assume that for every *finite* subset $J \subset I$ we have $\bigcap_{j \in J} (G \setminus K_j) \neq \emptyset.$   *Question*: Is it true that $\bigcap_{i \in I} (G \setminus K_i) \neq \emptyset$?

*Examples*: for $G = \mathbb{Z}$ and $H = \{0\}$, the ACC is satisfied, the coatoms are $(p\mathbb{Z})_{p \in \mathbb{P}}$ and $\bigcap_{p \in \mathbb{P}} (\mathbb{Z} \setminus p\mathbb{Z}) = \{-1,1 \} \neq \emptyset$. For $G = \mathbb{Z} \rtimes C_2$ and $H = C_2$, it works as well.   
Any other example (with $H$ core-free) is welcome!


  [1]: https://en.wikipedia.org/wiki/Ascending_chain_condition
  [2]: https://en.wikipedia.org/wiki/Atom_(order_theory)#Coatoms