On the image of the total power operation for Burnside rings

We prove that the image of the total power operation for Burnside rings $A(G)\to A(G\wr {\mathrm{\Sigma }}_n)$ lies inside a relatively small, combinatorial subring $\mathring{A}(G,n)\subseteq A(G\wr {\mathrm{\Sigma }}_n)$. As $n$ varies, the subrings $\mathring{A}(G,n)$ assemble into a commutative graded ring $\mathring{A}(G)$ with a universal property: $\mathring{A}(G)$ carries the universal family of power operations out of $A(G)$. We construct character maps for $\mathring{A}(G,n)$ and give a formula for the character of the total power operation. Using $\mathring{A}(G)$, we extend the Frobenius–Wielandt homomorphism of Dress–Siebeneicher–Yoshida to wreath products compatibly with the total power operation. Finally, we prove a generalization of Burnside’s orbit counting lemma that describes the transfer map $A(G\wr {\mathrm{\Sigma }}_n)\to A({\mathrm{\Sigma }}_n)$ on the subring $\mathring{A}(G,n)$.