Finite groups in which elements of the same order outside the center are   conjugate
  In this paper, we prove that if $G$ is a finite group in which elements of the same order outside the center are conjugate, then $G$ is abelian or is isomorphic to $S_3$, the symmetry group of degree 3. 
Guohua Qian, Wujie Shi, Xingzhong You