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We compute the $G$-equivariant homotopy type of the real topological cyclichomology of spherical group rings with anti-involution induced by takinginverses in the group, where $G$ denotes the group$Gal(\mathbb{C}/\mathbb{R})$. The real topological Hochschild homology of aspherical group ring $\mathbb{S}[\Gamma]$, with anti-involution as described,is an $O(2)$-cyclotomic spectrum and we construct a map commuting with thecyclotomic structures from the $O(2)$-equivariant suspension spectrum of thedihedral bar construction on $\Gamma$ to the real topological Hochschildhomology of $\mathbb{S}[\Gamma]$, which induce isomorphisms on $C_{p^n}$- and$D_{p^n}$-homotopy groups for all $n\in \mathbb{Z}$ and all primes $p$. Here$C_{p^n}$ is the cyclic group of order $p^n$ and $D_{p^n}$ is the dihedralgroup of order $2p^n$. Finally, we compute the $G$-equivariant homotopy type ofthe real topological cyclic homology of $\mathbb{S}[\Gamma]$ at a prime $p$.

Real topological cyclic homology of spherical group rings