I am trying to understand the classical method of counting classes from Burnside's old book (Note E) (also clarified a bit by Vera-Lopez, Conjugacy classes in finite solvable groups, 1984) : $G$ is a finite group with $N\unlhd G$ of index $p$ prime. If $g \in G \setminus N$ and $C_N(n_1),\dotsc,C_N(n_s)$ are classes fixed by the automorphism $\psi:a \mapsto g^{-1}ag$ on $N$, then all these classes are also $G$-classes. The remaining $r(N)-s$ classes form disjoint orbits of size $p$ by the outer automorphism induced by $\psi$. Hence $p$ of them fuse to give $(r(N)-s)/p$ classes. This counts $s + (r(N)-s)/p$ $G$-classes.
Now the remaining $G$-classes come from the cosets $g^jN (1 \leq j \leq p-1)$. There is a $G$-action on these cosets by conjugation inducing the following equation : $u_x |N| = \sum_{m \in N} \theta_x(m) = |S_x|$, where $S_x = \{ (w,n)\in xN \times N: n^{-1}wn = w \}$, $\theta_x(m)$ counts number of $w$ fixed by $m$. Here $u_x$ is supposed to mean number of orbits of $(xN, N)$ which I don't understand. Which action is it referring to?
