This post is a sequel of Generalization of the fundamental theorem of cyclic groups
Let $G$ be a finite group then the fundamental theorem of cyclic groups can be formulated as follows:
Theorem: $G$ is cyclic iff it admits no two different subgroups with the same order.
proof: see here p44 together with Lagrange theorem. $\square$
We ask about a generalization of one side of this statement for an inclusion $(H \subset G)$ of finite groups:
Definition: $G$ is called $H$-cyclic if $\exists g \in G$ such that $\langle H,g \rangle = G$.
Question: Is it true that if $(H \subset G)$ admits no two different intermediate subgroups $H \subset K \subset G$ with the same order, then $G$ is $H$-cyclic?
Remark: It is checked (by GAP) for $[G:H] \le 31$ and this answer of John Shareshian gives examples beyond. The converse is false because the inclusion $(S_2 \times S_2 \subset S_3 \times S_3)$ is a counter-example.