is easy to see that $\phi$ induces a bijection between cells of $\Pi_G^*$ and of $\Pi_H^*$. Apparently, if ${\mathrm{Aut} \hspace{0.4mm} G}$ is trivial, [*i.e.,*]{} there is only one permutation, the identity, in ${\mathrm{Aut} \hspace{0.4mm} G}$, then the bijection from $\Pi_G^*$ to $\Pi_H^*$ is actually equal to $\phi$. So let us consider more interesting cases and assume ${\mathrm{Aut} \hspace{0.4mm} G}$ possesses at least one non-trivial orbit.

We take a vertex $u_1$ from a non-trivial orbit of ${\mathrm{Aut} \hspace{0.4mm} G}$. Then there is exactly one vertex $v_1$ of $H$ corresponding to $u_1$ through $\phi$, and accordingly $\phi$ induces a bijection between cells of $\Pi_{u_1}^*$ and of $\Pi_{v_1}^*$, where $\Pi_{u_1}^*$ and $\Pi_{v_1}^*$ are two partitions of $V(G)$ and $V(H)$, respectively, consisting of orbits of $( {\mathrm{Aut} \hspace{0.4mm} G} )_{u_1}$ and of $( {\mathrm{Aut} \hspace{0.4mm} H} )_{v_1}$, and $( {\mathrm{Aut} \hspace{0.4mm} G} )_{u_1}$ stands for the subgroup of ${\mathrm{Aut} \hspace{0.4mm} G}$ which is defined as $\{ \gamma \in {\mathrm{Aut} \hspace{0.4mm} G} : \gamma \hspace{0.5mm} u_1 = u_1 \}$ and called the [*stabilizer of $u_1$ in ${\mathrm{Aut} \hspace{0.4mm} G}$.*]{} Moreover, if $( {\mathrm{Aut} \hspace{0.4mm} G} )_{u_1}$ is non-trivial, we could choose another vertex $u_2$ from a non-trivial orbit of $( {\mathrm{Aut} \hspace{0.4mm} G} )_{u_1}$. Then 