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

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

Under what conditions on $G$ is the following statement true:

Product of two elements $a$, $b$ in a conjugacy class $C$ of central involutions belongs to $C$ if and only if $a$ and $b$ commute.

My guess is that if $G$ has a unique conjugacy class of centralizers then this result is true. Though I don't have a proof of it (yet). Is that condition sufficient? If yes, then is it also necessary?