Michio Suzuki showed that every finite, simple, non-abelian, CA-group is of even order.
A textbook exposition of the classification of finite CA-groups is given as example 1 and 2 in (Suzuki 1986, pp. 291–305).
A more detailed description of the Frobenius groups appearing is included in (Wu 1998), where it is shown that a finite, solvable CA-group is a semidirect product of an abelian group and a fixed-point-free automorphism, and that conversely every such semidirect product is a finite, solvable CA-group.
Wu also extended the classification of Suzuki et al. to locally finite groups.
Wu also observes that Tarski monsters are obvious examples of infinite simple CA-groups.