Anticommutativity of Symmetric Elements under Generalized Oriented Involutions

Let $R$ be a ring with $char(R)\mathrm{\ne }2$ whose unit group are denoted by $\mathcal{U}(R)$, $G$ a group with involution $\ast$, and $\sigma :G\to \mathcal{U}(R)$ a nontrivial group homomorphism, with $ker
\sigma=N$,satisfying$xx^\in N$forall$x\in G$\mathrm{.}Let$RG$ be the group ring of $G$ over $R$ and define the involution $\sigma \ast$ in $RG$ by $\left( \sum_{x\in G}\alpha_xx\right)^{\sigma}=\sum_{x\in G}\sigma(x)\alpha_xx^*$. In this paper, we will classify the group rings $RG$ such that $\mathcal{S}$ is anticommutative, where $\mathcal{S}$ is the largest subset of $(RG)^+=\left{ \alpha\in RG: \alpha^{\sigma *}=\alpha\right}$ that can satisfy anticommutativity under $char(R)\mathrm{\ne }2$.