Characteristic subgroups in locally finite groups

Let G be a locally finite group having a subgroup N of finite index n that possesses a normal seriesN=N0⩾N1⩾⋯⩾Nk=1N=N0⩾N1⩾⋯⩾Nk=1 each of whose quotients Ni/Ni+1Ni/Ni+1 is either locally nilpotent or satisfies an outer commutator law wi≡1wi≡1. We show that G contains a characteristic subgroup H of finite index that has a characteristic series with the same properties. Moreover, the index of H in G is bounded by a function depending only on n, k   and the weight of the wiwi.