The study of homology groups =s(II, A), 'S a variety, II a group in ii, and A a suitable Il-module, is continued. A 'Tor' is constructed which gives a better (but imperfect) approximation to these groups than a Tor previously con- sidered, ii (II, Z) is calculated for various varieties 3i and groups II. Introduction. We continue the study of homology groups 33 (IT, A), where EI is a group in the variety 3} and A is a suitable Fl-module, as in (2ll, ^251 and (31), here after referred to as (H i), (H II) and (H III). These homology groups were compared with a certain Tor in (H i) and (H II), and it was shown that the two theories do not always agree in dimension 1. In § 1 we introduce a different Tor which gives a better approximation to 3} (n, A); this theory arises from the consideration of two-sided modules as in the Hochschild theory, and was prompted by a remark of Barry Mitchell's. In §2,3} (n, Z) is calculated approximately for 3} the variety of metabelian groups that are nilpotent of class c and FI a finitely generated abelian group. The homological techniques are the same as those used in (H III) to calculate 3} (n, Z) for 5} the variety of all nilpotent groups of class c and II as above, though the group theory here is harder. Comparing the results in the two cases we deduce that neither Tor mentioned above always agrees with 3) (II, A) if 72 = 2, even when A is a module with trivial action. In §3 we repeat the above calculations, for 3) the two varieties of all metabelian and all centre- by-metabelian groups. A refinement in the homology enables the calculations to be exact in these cases. The variety of metabelian groups is particularly inter- esting in that in this case it follows from a theorem of Grace Orzech that 33 (n, A), which can be calculated from 3} (JJ, Z) in the case of trivial action, may be interpreted as an obstruction group much as in the classical theory. The principal conventions and notations of (H I), (H II) and (H III) will re- main in force. A finitely generated abelian group will be said to be of rank r and type (s; ?2., - • • , 72 ) if it is the direct product of s infinite cyclic groups, and t finite cyclic groups, one of order 72. for each i, where n . divides 72 for 1 < 2 < t, and r = s + t. A cyclic group generated by the element a will be written