Nilpotent cohomology classes and some examples of groups

In view of its importance for the study of idempotents in group rings, a certain class image of groups, defined by means of a cohomological condition, was introduced by Emmanouil (Invent. Math. 132 (1998) 307–330). In the present paper, we establish the limitation of that class by constructing explicit examples of groups that do not satisfy that cohomological condition.