Dimension subgroups and Lie dimension subgroups are known to satisfy a `universal coefficient decomposition', {\it i.e.} their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the `universal' coefficient rings given by the cyclic groups of infinite and prime power order. Here this fact is generalized to much more general types of induced subgroups, notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary $N$-series, as well as certain common generalisations of these which occur in the study of the former. This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi, Parmenter and Seghal), to all additive subgroups of group rings. This is possible by using homological instead of ring theoretical methods.