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 $x{x}^{\sigma }=x^{\sigma }x$ for all $x={\sum }_{g\in G}{\alpha }_{g}g\in KG$, where \quad $x\mapsto x^\sigma=~\sum_{g\in G}\alpha_gf(g)\sigma(g)$isaninvolutionof$KG$;here$f: G\to U(K)$ is a homomorphism and $\sigma$ is an anti-automorphism of order two of $G$.