Iterated semidirect-product conjecture for braid-orbit permutation groups

Let $G>I$ be a finite group with $Z(G)=I$. Let $(C_1,C_2,C_3,C_4)$ be a class vector of length $4$ of $G$, let $Z\subseteq\Sigma^i_4(C_1,C_2,C_3,C_4)$ be a $B_4$-orbit with $q=\lvert Z\rvert>0$, and let $\rho_4:B_4\to S_q$ be the induced permutation representation of the action of $B_4$ on $Z$.

Iterated semidirect-product conjecture.

$$\rho_4(B_4)\cong C_{n_l}^{k_l}\rtimes(\cdots\rtimes(C_{n_2}^{k_2}\rtimes(C_{n_1}^{k_1}\rtimes G_m))\cdots)\leq S_q,$$

with $G_m\in\{A_m,S_m\}$, $q\mid m\cdot n_1\cdot\dots\cdot n_l$, $m\leq q$, $l\in\mathbb{N}_0$, $n_1,\dots,n_l\in\{2,3\}$, $k_1,\dots,k_l\in\mathbb{N}$, and $k_1=m-1$.

The conjecture proposes a restricted structure for the permutation groups arising from braid-group actions on these finite orbits. The supplied text motivates it through computations of examples, but provides no resolution.