On symmetric units in group algebras

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K. The anti-automorphism g → g^-1 of G can be extended linearly to an anti-automorphism a → a* of KG. Let S_*(KG) = {x ∈ U(KG) | x* = x} be the set of all symmetric units of U(KG). We consider the following question: for which groups G and commutative rings K it is true that S_*(KG) is a subgroup in U(KG). We answer this question when either a) G is torsion and K is a commutative G-favourable integral domain of characteristic p > 0 or b) G is non-torsion nilpotent group and KG is semiprime.