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We continue our study of the genus-$0$ permutation-equivariant quantumK-theory of the target $X=pt$, and completely determine the "big J-function" ofthis theory. The computation is based on the application of Lefschetz' fixedpoint formula to the action of $S_n$ on $\overline{M}_{0,n+1}$. It is aninstance of the general "adelic characterization" (which we state at the endwith reference to arXiv:1106.3136) of quantum K-theory for any target $X$ interms of quantum cohomology theory. Yet, some simplifications of non-conceptualnature occur in this example, making it a lucid illustration to the generaltheory.

Permutation-equivariant quantum K-theory III. Lefschetz' formula on $\overline{M}_{0,n}/S_n$ and adelic characterization