Burnside's Lemma states that, given a set $X$ acted on by a group $G$,
where $|X/G|$ is the number of orbits of the action, and $|X^g|$ is the number of fixed points of $g$. In other words, the number of orbits is equal to the average number of fixed points of an element of $G$.
Is there any way in which the fixed points of an element $g$ can be thought of as orbits? I had wondered aloud on my recent question here how (or if) Burnside's Lemma can be interpreted as having the same kind of object on both sides, so as to be a "true" average theorem, e.g.
"number of orbits = average over $g\in G$ of (number of orbits satisfying (something to do with $g$))"
"number of orbits = average over $g\in G$ of (number of orbits of some new action which depends on $g$)"
Since Qiaochu stated the comments to my question that he suspects Burnside's Lemma can be categorified, and that this may be related, I have also added that tag.