Groups with boundedly Chernikov conjugacy classes

A relevant theorem of B.H. Neumann states that if a group G has boundedly finite conjugacy classes, then its commutator subgroup G0 is finite. This result has recently been generalized by Detomi, Donadze, Morigi and Shumyatsky, who have proved that if the orbits of a group G under the action of G' by conjugation are boundedly finite, then the third term of the lower central series has boundely finite order. The aim of this paper is to prove a corresponding statement when boundedly finite orbits are replaced by boundedly Chernikov orbits.