Finite group having exactly one conjugacy class of involutions need not have order less than the cube of the order of the centralizer of involution

Statement
It is possible to have the following: $$G$$ is a finite group having exactly one conjugacy class of involutions, but the order of $$G$$ is greater than the cube of the order of the centralizer of any involution.

Opposite facts

 * Finite group having at least two conjugacy classes of involutions has order less than the cube of the maximum of orders of centralizers of involutions

Example of the dihedral group
Suppose $$n$$ is an odd integer greater than $$3$$. Consider the dihedral group $$D_{2n}$$. This has exactly one conjugacy class of involutions of size $$n$$ and all the centralizers are of order two. However, the order of the group is $$2n$$, which is greater than $$2^3 = 8$$.