On Fox and augmentation quotients of semidirect products

Let G be a group which is the semidirect product of a normal subgroup N and some subgroup T. Let In(G), n⩾1, denote the powers of the augmentation ideal I(G) of the group ring Z(G). Using homological methods the groups Qn(G,H)=In−1(G)I(H)/In(G)I(H), H=G,N,T, are functorially expressed in terms of enveloping algebras of certain Lie rings associated with N and T, in the following cases: for n⩽4 and arbitrary G, N, T (except from one direct summand of Q4(G,N)), and for all n⩾2 if certain filtration quotients of N and T are torsion-free.