# Do small subsets of $S_n$ subgroups cover almost all permutation configurations of $S_n$?

Given integer $$m\in[1,n]$$ fix a set $$\mathcal T$$ of permutations in $$S_n$$. Then there are subgroups $$G_1,\dots,G_m$$ of $$S_n$$ so that $$\mathcal T$$ is covered by cosets of $$G_1,\dots,G_m$$.

1. For every $$n\in\mathbb N$$ do we have an $$m=O(poly(n))$$ and an $$m'=O(poly(n))$$ such that there are elements $$g_1,\dots,g_m\in S_n$$ and subgroups of $$G_1,\dots,G_m$$ of $$S_n$$ such that for at least $$\frac1{poly(n)}$$ of all such $$\mathcal T$$ there is a subset $$\mathcal D\subseteq\{1,\dots,m\}$$ with $$\mathcal T\subseteq\cup_{i\in\mathcal D}g_iG_i$$ $$\big(|\mathcal T|-\sum_{i\in\mathcal D}|G_i\big)^2\leq m'$$ holds?

2. Does $$m=O(poly(n))$$ hold if $$m'=0$$ for at least $$\frac1{poly(n)}$$ of all $$\mathcal T$$ (giving $$\mathcal T=\cup_{i\in\mathcal D}g_iG_i$$)?