Let $G$ be a finite group acting on a set $X$, and let $S\subseteq G$ be a union of conjugacy classes. Then I believe I can prove:

$$ \sum_{[x]\in X/G} \frac{|G_x \cap S|}{|G_x|} = \sum_{g\in S} \frac{|X^g|}{|G|} $$

where $G_x$ is the stabilizer of $x\in X$ and $X^g$ is the fixed-point set of $g\in G$. The assumption on $S$ makes ${|G_x \cap S|}$ depend only on the orbit $[x]$ of $x$.

When $S=G$, this reduces to the orbit-counting theorem. Does the general form have a name? Or is it a special case of something that has a name? Is there somewhere I can cite for it?