such that $\gamma_{j} B_{i_j} \neq B_{i_j}$ and $\gamma_{j} B_{i_{j+1}} = B_{i_{j+1}}$.

\[Thm-Equation-OrbitsOfAutG\] $$\mathbf{R}_{\Pi^*_G} V_{\lambda}^{ G / \Pi^*_G } 
= 
\left( \bigcap_{t\in T} \mathbf{R}_{\Pi_t^*} V_{\lambda}^{ G / \Pi_t^* } \right) 
\bigcap 
\left( \bigcap_{j=1}^r V_1^{ \gamma_j} \right)_{\textstyle .}$$

Accordingly, in order to determine the partition $\Pi^*_G$, we only need to have one partition $\Pi^*_t$ and a group of permutations $\gamma_1,\ldots,\gamma_{_{\footnotesize l(T)}}$ in $\mathfrak{G}$, where $t$ is an element of a non-trivial $\mathfrak{G}$ orbit $T$, which is composed of $t_1=t,t_2,\ldots,t_{_{\footnotesize l(T)}}$, and $\gamma_i\hspace{0.4mm} t = t_i$, $i=1,\ldots,l(T)$. Similarly, in order to determine $\Pi^*_t$, we only need to know one partition $\Pi^*_{t,u}$, which is composed of orbits of the stabilizer $\left( \mathfrak{G}_t \right)_u$, abbreviated to $\mathfrak{G}_{t,u}$, and a group of permutations $\delta_1,\ldots,\delta_{l(R)}$ in $\mathfrak{G}_t$ such that $u$ belongs to some non-trivial orbit $R$ of $\mathfrak{G}_t$, which consists of elements $u_1=u,u_2,\ldots,u_{l(R)}$, and $\delta_i\hspace{0.4mm} u = u_i$, $i=1,\ldots,l(R)$. Apparently, we can repeat this process until the partition consisting of orbits of the last stabilizer, which fixes a sequence of vertices of $G$, is made up of trivial cells only, [*i.e.,*]{} the final partition is equal to $\{\{v\} : v \in [n] \}$.

We call a sequence of vertices $u_1,\ldots,u_s$ a fastening sequence of $\mathfrak{G}$ if $u_1$ 