Groups with finitary classes of conjugate elements

A class of conjugate elements of a group is called finitary if the action of group by conjugation induces a group of finitary permutations of this class. A group with finitary classes of conjugate elements will be called a ФC-group. Some characterizations of ФC-groups in the class of all groups are obtained. It is also shown for every ФC-group that either it is an FC-group, i.e., a group with finite classes of conjugate elements, or its structure is close to the structure of a totally imprimitive group of finitary permutations.