On the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}\mathrm{/}4$

We study the homotopy groups of the geometric fixed points of the real topological cyclic homology of $\mathbb{Z}\mathrm{/}4$. We relate these groups to the values of the non-abelian derived functors of the functor $M\mapsto (M{\otimes }_{\mathbb{Z}\mathrm{/}4}M{)}^{C_2}$ at the $\mathbb{Z}\mathrm{/}4$-module $\mathbb{Z}\mathrm{/}2$, which we precisely calculate with computer assistance up to degree $6$, and calculate in general up to slight remaining ambiguity. Using these results we compute ${\pi }_i(\mathrm{T}\mathrm{C}\mathrm{R}(\mathbb{Z}\mathrm{/}4{)}^{\phi \mathbb{Z}\mathrm{/}2})$ exactly for $i\le 1$, up to an extension problem for $2\le i\le 5$, and describe the asymptotic growth of this group for large $i$. A consequence of these computations is that there exists some $0\le i\le 5$ such that the canonical map comparing the genuine symmetric and symmetric $L$-theory spectra of $\mathbb{Z}\mathrm{/}4$ is not an isomorphism on degree $i$ homotopy, and moreover this comparison map is never an isomorphism on homotopy in sufficiently large degrees.