A note on nilpotent-by-Cernikov groups

In this note we prove that a locally graded group G in which all proper subgroups are (nilpotent of class not exceeding n)-by-Cernikov, is itself (nilpotent of class not exceeding n)-by-Cernikov. As a preparatory result that is used for the proof of the former statement in the case of a periodic group, we also prove that a group G, containing a nilpotent of class n subgroup of finite index, also contains a characteristic subgroup of finite index that is nilpotent of class not exceeding n.