**Question:** does there exist a strictly ascending sequence of finite groups
$G_0<G_1<G_2<\dots $ such that for every $i \in \mathbb{N}$ there is $a_i \in G_{i+3}$ and the following two conditions are satisfied:

(i) $a_i$ centralizes $G_{i-1}$ in $G_{i+3}$;

(ii) $G_{i+3}=\langle G_i, a_i G_i a_i^{-1} \rangle$.

*Remark:* taking $G_i=Sym(d_i)$ for some increasing sequence of integers $(d_i)$ with the evident embedding $Sym(d_i) \hookrightarrow Sym(d_{i+1})$ does not seem to work.

6more comments