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Definition: An inclusion of finite groups $(A \subset B)$ is relatively cyclic if $\exists b \in B$ such that $\langle A,b \rangle = B$.

Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$.

Remark: The equivalence class of $(A \subset B)$ is completely characterized by the conjugacy class of a transitive permutation group $G$ such that $(A \subset B) \sim (G_1 \subset G)$.

Let $p_n$ be the ratio of index $\le n$ relatively cyclic inclusions (up to equivalence).
Let $\alpha_n = \liminf_{r>n} (p_r)$ and $\alpha = \lim_{n \to \infty} (\alpha_n) $.

Question: what's the value of $\alpha$ (or at least good bounds)?
Bonus: what's the asymptotic analysis of the sequence $(\alpha_n-\alpha)$?

GAP computation:
Up to equivalence, there are $15341$ inclusions of finite groups $(A \subset B)$ of index $[B : A] \le 30$ and $\vert B \vert \le 10^4$, and among them, $12671$ are relatively cyclic (more than $80 \%$). Note that at index exactly $16$ (and without order restrictions) less than $70 \%$ are relatively cyclic.

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