I am trying to understand group theoretic algorithms for graph isomorphism problem. In page number 64 and 65 of this book chapter, the author has bounded the index of the concerned automorphism groups (lemma 2 and 3). I really did not understand how does this statement - "if $\pi$ and $\psi \in G^{(r+j-1)}$ such that $\pi\psi^{(-1)}$ stabilizes every vertex in $C_j$ , then $\pi$ and $\psi$ lie in the same right coset of $G^{(r+j)}$"- leads to the bound in lemma 3. I found the argument in proof of lemma 2 even more confusing. Is there some standard techniques to bound the number of cosets being employed here?
Here the vertex set of graph is partitioned into $s$ colour classes each with maximum size $k$, which are to be preserved by the isomorphism. They create $\binom{s}{2} +1$ induced graphs, each having vertex set $V$, the vertex set of original graph, and starting from empty set edge set are build inductively, where at each step, edges having one vertex in class $i$ and other in class $j$ are added.