is in the subspace $\mathbf{R}_{\Pi} V_{\lambda}^{ G / \Pi }$, where $V_{\lambda}^{ G / \Pi }$ is the eigenspace of ${\mathbf{A}( G / \Pi )}$ corresponding to $\lambda$, and thus $$\begin{aligned}
{\mathrm{proj}\big[ V_{\lambda} \big]}( \pmb{R}_C ) 
& = {\mathrm{proj}\big[  \mathbf{R}_{\Pi} V_{\lambda}^{ G / \Pi }  \big]}( \pmb{R}_C )  \\
& = \sum_{c\in C} {\mathrm{proj}\big[  \mathbf{R}_{\Pi} V_{\lambda}^{ G / \Pi }  \big]}( \pmb{e}_c )  \\
& = |C| \cdot {\mathrm{proj}\big[  \mathbf{R}_{\Pi} V_{\lambda}^{ G / \Pi }  \big]}( \pmb{e}_c ).\end{aligned}$$ On the other hand, $\pmb{R}_C = \sum_{ \lambda \in {\mathrm{spec} \hspace{0.4mm}  {\mathbf{A}( G )} } } {\mathrm{proj}\big[ V_{\lambda} \big]}( \pmb{R}_C )$. Therefore, $$\label{Equ-OrbitsAutG-Equitable}
\frac{1}{ |C| } \cdot \pmb{R}_C = 
\sum_{ \lambda \in {\mathrm{spec} \hspace{0.4mm}  {\mathbf{A}( G )} } } {\mathrm{proj}\big[  \mathbf{R}_{\Pi} V_{\lambda}^{ G / \Pi }  \big]}( \pmb{e}_c ), ~~~~ \forall c \in C.$$ This relation reveals that in order to determine the partition $\Pi_G^*$, we only need to work out those subspaces $\mathbf{R}_{\Pi_G^*} V_{\lambda}^{ G / \Pi_G^* }$ for each $\lambda \in {\mathrm{spec} \hspace{0.4mm}  {\mathbf{A}( G )} }$.

Before showing how to obtain $\mathbf{R}_{\Pi_G^*} V_{\lambda}^{ G / \Pi_G^* }$ without knowing the partition $\Pi_G^*$, we first introduce two kinds of subspaces $V_{\lambda} \langle u \rangle$ and $V_{1}^{\gamma}$ relevant to subgroups of 