On separable group rings

Let $G$ be a finite non-abelian group $R$ a ring with 1, and $\overline G$ the inner automorphism group of the group ring $RG$ over $R$ induced by the elements of $G$. Then three main results are shown for the separable group ring $RG$ over $R$: (i) $RG$ is not a Galois extension of $(RG)^{\overline G}$ with Galois group $\overline G$ when the order of $G$ is invertible in $R$, (ii) an equivalent condition for the Galois map from the subgroups $H$ of $G$ to $(RG)^H$ by the conjugate action of elements in $H$ on $RG$ is given to be one-to-one and for a separable subalgebra of $RG$ having a preimage, respectively, and (iii) the Galois map is not an onto map.