intersection of all $2$-sylow in $S_n$

Question

I am currently stuck on an exercise that asks to determine explicitly the subgroup $O_2(S_n)<S_n$ given by the intersection of all Sylow $2$-subgroups, for $n\geq 4$. I already proved that $O_2(S_n)$ is the unique largest normal $p$-subgroup of $S_n$, but I do not see how to use this to solve the exercise. Can somebody please give me a hint?