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An involution $a$ of a group $G$ is called central if there exists a sylow $2$-subgroup $H$ of $G$ such that $a \in C_G(H)$.

Clearly if an involution is central then its every conjugate is also central.

  1. If $C$ is a conjugacy class of involutions and $a$, $b$ are two distinct members of $C$ that commute with each other then is $ab$ also a member of $C$?

  2. Can we classify all finite groups (or finite simple groups) for which the number of conjugacy classes of central involutions is $1$?

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I don't know how your definition of central involution is related to the first question; anyway, here is a counterexample of your first proposition:

Let $G = \langle (3, 4), (1, 3)(2, 4) \rangle \cong D_8$. Set $a = (1, 3)(2, 4)$ and $b = (1, 4)(2, 3)$. Then $a^2 = b^2 = \operatorname{id}$, $ab = ba$ and $C = \{ a, b \}$ is a conjugacy class of $G$, but $ab = (1, 2)(3, 4) \not\in C$.

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