Homology operations in the topological cyclic homology of a point

We consider the commutative S-algebra given by the topological cyclic homology of a point. The induced Dyer-Lashof operations in mod p homology are shown to be nontrivial for p = 2, and an explicit formula is given. As a part of the calculation, we are led to compare the fixed point spectrum S(G) of the sphere spectrum and the algebraic K-theory spectrum of finite G-sets, as structured ring spectra.