I'm trying to follow a sketch proof about the abstract characterization of $S_5$, by Walter Feit.
Suppose $G$ is a finite group with exactly two conjugacy classes of involutions, with $u_1$ and $u_2$ being representatives. Suppose $C_1=C(u_1)\simeq \langle u_1\rangle\times S_3$ and $C_2=C(u_2)$ be a dihedral group of order $8$. The eventual result is that $G\simeq S_5$. Also, $C(u)$ denotes the centralizer of $u$ in $G$.
I don't understand the observation some involution is in the center of a Sylow subgroup, and that $C_2$ is a Sylow $2$-subgroup. I do know that $C_2$ is contained in a Sylow $2$-subgroup at least, from the Sylow theorems, but without knowing the actual order of $G$, I don't see why it necessarily a Sylow subgroup itself.