The Riemann-Hurwiz formula and a special type of group action

Suppose is some finite group acting on the set , with the special condition that each element of fixes points of . Then we can write . (Counting the non-identity elements of in two different ways.)

Now this formula strongly resembles the Riemann-Hurwiz formula, which relates the Euler characteristics of two Riemann surfaces , one of which is a degree ramified covering of the other : we then have have , where is the ramification index at .

Are these formulas related directly?