On symmetric units in group algebras

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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 g1 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.