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 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 $[H,G]$ be an ACC interval of groups and $(K_i)_{i \in I}$ its coatoms, i.e. the maximal elements in $[H,G)$. Assume that $I$ is an infinite countable set, and 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!

Application: Generalization of a theorem of Øystein Ore in group theory: the infinite case