How to find largest supergroup in polynomial time?
Yes, this can be solved in polynomial time (and, I think, even in $\mathsf{NC}$).

First, it's easy to reduce to the case of transitive $G$.

Next, find the minimal blocks of $G$. (See e.g. Section 3.6(B) of Dixon &amp; Mortimer.) This can be done by choosing a point $x \in [n]$, and for each $y \in [n]$, $y \neq x$, computing the (undirected) connected components of the orbital digraph generated by $(x,y)$. Each such connected component is a block; take the smallest ones.

If there are $b$ blocks each of size $s$, then the desired group is the wreath product $S_b \wr S_s = (S_b)^s \rtimes S_s$ in its natural action on $[n]$.