Group rings in which every element is uniquely the sum of a unit and an idempotent

A ring R is called clean if every element is the sum of an idempotent and a unit, and R is called uniquely clean if this representation is unique. These rings are related to the boolean rings: R is uniquely clean if and only if R/J(R) is boolean, idempotents lift modulo J(R), and idempotents in R are central. It is shown that if the group ring RG is uniquely clean then R is uniquely clean and G is a 2-group. The converse holds if G is locally finite (in particular if G is solvable).