Large subgroups of $S_n$ without large symmetric or alternating subgroups

Question

I'm interested in determining the existence of a permutation group $G\subseteq S_n$ of the following form.

1. $G$ is large. Meaning that $G$ have at least $n!/2^{o(n)}$ elements. Equivalently, their index is at most $2^{o(n)}$.
2. For some fixed $d$, no subgroup $H$ of $G$ is isomorphic to $S_d$ or to $A_d$.

Question 1: How small can $d$ be? For instance, is there a subgroup $G$ of $S_n$ of index $2^{o(n)}$ such that no subgroup of $G$ is isomorphic to $S_{\sqrt{n}}$ or $A_{\sqrt{n}}$?

The second question is whether there are Ramsey-theoretic results relating symmetric groups of size $d$ with large subgroups of $S_n$.

Question 2: Are there functions $f,g:\mathbb{N}\rightarrow \mathbb{N}$ such that for every $d<f(n)$, any subgroup $G$ of $S_n$ of order at least $g(d)$ contains a subgroup $H$ isomorphic to $S_d$ (or to $A_n$)?

Answer 1

Essentially the only subgroups of $S_n$ with subexponential index have an orbit of size greater than $n/2$ on which the induced action contains the alternating group. So it looks as though the answer to your Question 1 is yes, even with $n/2$ in place of $\sqrt{n}$.

This follows from Theorem 5.2B of "Permutation Groups" by Dixon and Mortimer, applied with $r = \lfloor n/2 \rfloor$.

Let $n := |\Omega| \ge 5$, let $r$ be an integer with $1 \le r \le n/2$, and let $G \le S_\Omega$ have index less than $\binom{n}{r}$. Then one of the following holds:

(i) for some $\Delta \subset \Omega$ with $|\Delta|<r$, we have $A_{(\Delta)} \le G \le S_{\{\Delta\}}$;

(ii) $n=2m$ is even and $G$ is imprimitive with two blocks of size $m$ and $|S_\Omega:G| = \frac{1}{2}\binom{n}{m}$;

(iii) one of six specific exceptional cases holds (which I won't list here).

Here $A_{(\Delta)}$ denotes the pointwise stabilizer of $\Delta$ in $A_\Omega$, and $S_{\{\Delta\}}$ is the setwise stabilizer of $\Delta$ in $S_\Omega$.

Note that the the index in Case (ii) is greater than $2^m$.