A finite transitive permutation group $G$ can always be ``decomposed'' into primitive permutation groups, called its primitive components, although the decomposition is not unique. See Chapter 1 of *Finite Permutation Groups* by Wielandt.

Suppose $H$ is a primitive component of $G$, and $T$ is a composition factor of $H$ that is also a classical group $X(d,q)$. Here $X$ stands for $\mathrm{PSL}$, $\mathrm{PSU}$, etc. Then $T$ is a subquotient of a composition factor $T'$ of $G$. If $T'$ is an alternating group $\mathrm{Alt}_k$, is there any lower bound on its degree $k$, in terms of $d$ and $q$?

If $T$ is a subgroup of $T'$ then $T$ has a faithful permutation representation of degree $k$. In this case I think a lower bound of order $q^{\Theta(d)}$ is known. See:

Cooperstein, Bruce N. "Minimal degree for a permutation representation of a classical group." Israel J. Math. 30 (1978), no. 3, 213-235. MR 506701 DOI: 10.1007/BF02761072.

I wonder if a similar bound is possible if $T$ is only a subquotient of $T'$.

A related question: for $k\in\mathbb{N}^+$, denote by $\Gamma_k$ the family of finite groups whose nonabelian composition factors are all isomorphic to subgroups of $\mathrm{Sym}_k$. If $G$ is a group in $\Gamma_k$, are its primitive components also in $\Gamma_k$ (or $\Gamma_{k'}$ for some $k'$ depending on $k$)?