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) → A(GΣₙ) lies inside a relatively small, combinatorial subring A(G,n) ⊆ A(G Σₙ). As n varies, the subrings A(G,n) assemble into a commutative graded ring A(G) with a universal property: A(G) carries the universal family of power operations out of $A(G)$. We construct character maps for A(G,n) and give a formula for the character of the total power operation. Using 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 Σₙ) → A(Σₙ) on the subring A(G,n).