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

Suppose $\displaystyle G$ is some finite group acting on the set $\displaystyle A$, with the special condition that each element of $\displaystyle G$ fixes $\displaystyle n$ points of $\displaystyle A$. Then we can write $\displaystyle n(|G|-1) = \sum_{a \in A}{(S_G(a)-1)$. (Counting the non-identity elements of $\displaystyle G$ in two different ways.)

Now this formula strongly resembles the Riemann-Hurwiz formula, which relates the Euler characteristics of two Riemann surfaces $\displaystyle M_1\to M_2$, one of which is a degree $\displaystyle n$ ramified covering of the other : we then have have $\displaystyle \chi(M_1)=N\chi(M_2)-\sum_{p \in M_1} (r_{p} -1)$, where $\displaystyle r_p$ is the ramification index at $\displaystyle p$.

Are these formulas related directly?