  We prove that the image of the total power operation for Burnside rings $A(G)
\to A(G\wr\Sigma_n)$ lies inside a relatively small, combinatorial subring
$\mathring A(G,n) \subseteq A(G \wr \Sigma_n)$. 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 $A(G \wr \Sigma_n) \to A(\Sigma_n)$ on
the subring $\mathring A(G,n)$.
