Solution
= Solution
A Sylow $2$-subgroup of $S_4$ has order $8$. For example,
$$
P=\langle(1234),(13)\rangle\cong D_8.
$$
It is the normalizer of $\langle(1234)\rangle$. The six $4$-cycles form three inverse pairs, and a dihedral <group> of order eight has a unique cyclic <subgroup> of order four. Hence the three Sylow $2$-subgroups are
$$
\boxed{
N_{S_4}(\langle(1234)\rangle),\quad
N_{S_4}(\langle(1243)\rangle),\quad
N_{S_4}(\langle(1324)\rangle)}.
$$