Difference of fixed points by subgroup action

If $G$ is a finite group and $X$ is a finite $G$-set, then we have a class equation which tells us $$|X|=|X^G|+\sum |G/G_x|$$ where $X^G$ is the set of $G$-fixed points, $G_x$ is the stabilizer of $x\in X$ and the sum is taken over representative elements of classes of non-singleton $G$-orbits.

My question, which arise from a proof in a paper which seems to use precisely this fact, is the following. If $H\leq G$ is a normal subgroup, are we able to write the cartinality of $X^H\setminus X^G$ as sum of cardinalities of $G/H$ orbits? If this is the case, over what that sum is taken?

I guess the result should descend by comparing the two class equations for $H$ and $G$, but I'm struggling to write that down.

Thanks in advance! Any help is really appreciated.

Solution 1:

The group $G/H$ acts on $ X^H\setminus X^G$ without fixed points, and so the equation you cited gives $$|X^H\setminus X^G| = \sum |(G/H)/(G/H)_x|$$ in this case where the sum is over representative elements of the $G/H$-orbits (which are the same as the $G$-orbits) of $X^H\setminus X^G$.