Real topological cyclic homology of spherical group rings

We compute the $G$-equivariant homotopy type of the real topological cyclic homology of spherical group rings with anti-involution induced by taking inverses in the group, where $G$ denotes the group $Gal(\mathbb{C}/\mathbb{R})$. The real topological Hochschild homology of a spherical group ring $\mathbb{S}[\Gamma]$, with anti-involution as described, is an $O(2)$-cyclotomic spectrum and we construct a map commuting with the cyclotomic structures from the $O(2)$-equivariant suspension spectrum of the dihedral bar construction on $\Gamma$ to the real topological Hochschild homology 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 dihedral group of order $2p^n$. Finally, we compute the $G$-equivariant homotopy type of the real topological cyclic homology of $\mathbb{S}[\Gamma]$ at a prime $p$.