  -   Given an isometry $\sigma : \Gamma \to \Gamma$ and a collection of isomorphisms $\Phi = \{ \varphi_u : G_u \to G_{\sigma(u)} \mid u \in V(\Gamma) \}$, the *local automorphism* $(\sigma, \Phi)$ 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 & \text{$\varphi_u(g)$ if $g \in G_u$} \end{array} \right.