This is a repost of a question I was trying to solve yesterday that got deleted. The question asked for a characterization of the subgroups $G$ of $S_n$ which when endowed with their natural action on $2^{[n]}$ satisfy that if $M$ and $N$ are equal-sized subsets of $[n]$ then there is a permutation $\varphi$ in $G$ so that $\varphi(M)=N$.
So you don't ask what I have tried later:
The group is clearly primitive. If $n!/|G|=k$ The order of the stabilizer of a subset $S$ of size $m$ is $\frac{(n-m)!(m)!}{k}$ (This is because we want the same orbits as when $S_n$ is acting an we use the orbit-stabilizer theorem)
There is an old result of Jordan that a transitive subgroup of $S_n$ that contains an element of prime order $p$ with $n/2 < p < n-2$ is equal to $A_n$ or $S_n$. This applies to groups satisfying your hypothesis provided that there exists such a prime, which is true for $n \ge 8$.