Let be a group and be a normal subgroup of . If the set is composed by consecutive integers, then is either nilpotent or a quasi-Frobenius group with abelian kernel and complements. This is a generalization of Theorem 2 of [A. Beltrán, M. J. Felipe and C. G. Shao, -divisibility of conjugacy class sizes and normal -complements, J. Group Theory 18 (2015) 133-141].