Normality in group rings

Let $KG$ be the group ring of a group $G$ over a commutative ring $K$ with unity. The rings $KG$ are described for which $xx^\sigma=x^\sigma x$ for all $x=\sum_{g\in G}\alpha_gg\in KG$, where $x\mapsto x^\sigma=\sum_{g\in G}\alpha_gf(g)\sigma(g)$ is an involution of $KG$; here $f\colon G\to U(K)$ is a homomorphism and $\sigma$ is an antiautomorphism of order two of $G$.