I generally see the class equation stated for the action of a group on itself by conjugation as follows where $C(G)$ is the centralizer of $G$:
$$|G|=|C(G)|+\sum{sizes\ of\ nontrivial\ conjugacy\ classes}$$
...or...
$$|G|=|C(G)|+\sum [G:C_G(x)]$$
It seems pretty clear that the more general statement also true. That is, if $G$ acts on a set $S$:
$$|S|=|Fix_G(S)|+\sum{sizes\ of\ nontrivial\ orbits}$$
...or...
$$|S|=|Fix_G(S)|+\sum [G:Stab_G(x)]$$
Pardon the poor notation. In both cases, I am summing over non-trivial conjugacy classes, and $Fix_G(S)$ indicates the set of points in $S$ fixed by all $g\in G$.
First, I am asking if my overall statement is correct.
Second, if the overall statement is correct, I was wondering if there is a reason why the conjugacy class statement seems to get more prominence than the general statement. Is the general statement not very useful?