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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 $a \in C_G(H)$, 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 distinct elements $a$, $b$ in a conjugacy class $C$ of central involutions belongs to $C$ if and only if $a$ and $b$ commute.

Clearly if $G$ has a unique conjugacy class of involutions then the result is true. What if $G$ has more than one conjugacy classes of involutions but precisely one which contains a central involution? Is the product of two distinct central involutions which commute again a central involution? Are there any characterisations of $G$ for which the above statement is always true?

As a follow up, can we classify all finite groups that contain a single conjugacy class of central involutions?

EDIT1: Derek Holt has given an example in $A_8$ showing that having a unique conjugacy class of central involutions is not sufficient.

EDIT2: Central involutions are the same thing as involutions that lie in the center of some sylow $2$-subgroup. Moreover, the number of conjugacy classes of central involutions is equal to the number of involutions in the center of a sylow $2$-subgroup.

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

Clearly if $G$ has a unique conjugacy class of involutions then the result is true. What if $G$ has more than one conjugacy classes of involutions but precisely one which contains a central involution? Is the product of two distinct central involutions which commute again a central involution? Are there any characterisations of $G$ for which the above statement is always true?

As a follow up, can we classify all finite groups that contain a single conjugacy class of central involutions?

EDIT1: Derek Holt has given an example in $A_8$ showing that having a unique conjugacy class of central involutions is not sufficient.

EDIT2: Central involutions are the same thing as involutions that lie in the center of some sylow $2$-subgroup. Moreover, the number of conjugacy classes of central involutions is at most the number of involutions in the center of a sylow $2$-subgroup.

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

Clearly if $G$ has a unique conjugacy class of involutions then the result is true. What if $G$ has more than one conjugacy classes of involutions but precisely one which contains a central involution? Is the product of two distinct central involutions which commute again a central involution? Are there any characterisations of $G$ for which the above statement is always true?

As a follow up, can we classify all finite groups that contain a single conjugacy class of central involutions?

EDIT: Derek Holt has given an example in $A_8$ showing that having a unique conjugacy class of central involutions is not sufficient.

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

Clearly if $G$ has a unique conjugacy class of involutions then the result is true. What if $G$ has more than one conjugacy classes of involutions but precisely one which contains a central involution? Is the product of two distinct central involutions which commute again a central involution? Are there any characterisations of $G$ for which the above statement is always true?

As a follow up, can we classify all finite groups that contain a single conjugacy class of central involutions?

EDIT1: Derek Holt has given an example in $A_8$ showing that having a unique conjugacy class of central involutions is not sufficient.

EDIT2: Central involutions are the same thing as involutions that lie in the center of some sylow $2$-subgroup. Moreover, the number of conjugacy classes of central involutions is equal to the number of involutions in the center of a sylow $2$-subgroup.

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

Clearly if $G$ has a unique conjugacy class of involutions then the result is true. What if $G$ has more than one conjugacy classes of involutions but precisely one which contains a central involution? Is the product of two distinct central involutions which commute again a central involution? Are there any characterisations of $G$ for which the above statement is always true?

As a follow up, can we classify all finite groups that contain a single conjugacy class of central involutions?

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

Clearly if $G$ has a unique conjugacy class of involutions then the result is true. What if $G$ has more than one conjugacy classes of involutions but precisely one which contains a central involution? Is the product of two distinct central involutions which commute again a central involution? Are there any characterisations of $G$ for which the above statement is always true?

As a follow up, can we classify all finite groups that contain a single conjugacy class of central involutions?

EDIT: Derek Holt has given an example in $A_8$ showing that having a unique conjugacy class of central involutions is not sufficient.