Permutations covered by subgroups?

Given integer $$m\in[1,n]$$ fix a set $$\mathcal T$$ of permutations in $$S_n$$. 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. My problem then is given $$\mathcal T$$ is there always an $$m=O(poly(n))$$ such that there are elements $$g_1,\dots,g_m\in S_n$$ and some subgroups of $$G_1,\dots,G_m$$ of $$S_n$$ such that

$$\mathcal T\subseteq\cup_{i=1}^mg_iG_i$$ $$(\sum_{i=1}^m|G_i|-|\mathcal T|)^2 where both $$m$$ and $$m’$$ are $$O(poly(n))$$.

1. If not what is the trade off between $$m$$ and $$m’$$?

2. Is it possible to get at least $$O(subexp(n))$$ for both?

3. If $$m’=0$$ is there always a minimum $$m$$ for all $$\mathcal T$$?