A group G is said to be a group with Cernikov conjugacy classes or a CC-group if it induces on the normal closure of each one of its elements a group of automorphisms which is a Cernikov group, that is, a finite extension of an abelian group satisfying the minimal condition on subgroups. This concept is a natural extension of that an FC-group, that is, a group in which every element has a finite number of conjugates. It is known that if G is an FC-group then the central factor G/Z(G) is periodic. This result does not hold for CC-groups and in this paper we study CC-groups G in which the central factor G/Z(G) is periodic, a finiteness condition which has a deep influence on the structure of the group G. In particular, we characterize those CC-groups as above that are FC-groups by imposing some additional conditions on their structure.