Groups with anticentral elements

We study finite groups $G$ with elements $g$ such that $\mid {\mathbf{C}}_G(g)\mid =\mid G:G^{\mathrm{\prime }}\mid$. (Such elements generalize fixed-point-free automorphisms of finite groups.) We show that these groups have a unique conjugacy class of nilpotent supplements for the commutator subgroup and, using the classification of finite simple groups, that these groups are solvable.