On torsion-by-nilpotent groups

Let C be a class of groups, closed under taking subgroups and quotients. We prove that if all metabelian groups of C are torsion-by-nilpotent, then all soluble groups of C are torsion-by-nilpotent. From that, we deduce the following consequence, similar to a well-known result of P. Hall (1958, Illinois J. Math.2, 787–801): if H is a normal subgroup of a group G such that H and G/H′ are (locally finite)-by-nilpotent, then G is (locally finite)-by-nilpotent. We give an example showing that this last statement is false when “(locally finite)-by-nilpotent” is replaced with “torsion-by-nilpotent.”