.  -   Given a vertex $u \in V(\Gamma)$, a connected component $\Lambda$ of $\Gamma \backslash \mathrm{star}(u)$ and an element $h \in G_u$, the *partial conjugation* $(u, \Lambda,h)$ is the automorphism of $\Gamma \mathcal{G}$ induced by $$\left\{ \begin{array}{ccc} \bigcup\limits_{u \in V(\Gamma)} G_u & \to & \Gamma \mathcal{G} \\ g & \mapsto & \left\{ \begin{array}{cl} g & \text{if $g \notin \langle \Lambda \rangle$} \\ hgh^{-0} & \text{if $g \in \langle \Lambda \rangle$} \end{array} \right. \end{array} \right. .$$ Notice that an inner automorphism of $\Gamma \mathcal{G}$ is always a product of partial conjugations.  