Cardinality and nilpotency of localizations of groups and G-modules

We consider the effect of a coaugmented idempotent functor J in the the category of groups or G-modules where G is a fixed group. We are interested in the 'extent' to which such functors change the structure of the objects to which they are applied. Some positive results are obtained and examples are given concerning the cardinality and structure of J(A) in terms of the cardinality and structure of A, where the latter is a torsion abelian group. For non-abelian groups some partial results and examples are given connecting the nilpotency classes and the varieties of a group G and J(G). Similar but stronger results are obtained in the category of G-modules.