Let $\Gamma = (\mathbf{V},\mathbf{H}, \mathbf{L},g, v,i,\ell)$ be a stable graph. Let $(\mathcal{C},p_1,\cdots,p_n)$ be a stable curve with the dual curve $\Gamma$ and let $(\tilde{\mathcal{C}}_v,(q_h)_{h\in \mathbf{H}(v)})_{v\in \mathbf{V}(\Gamma)}$ be the components of the normalization of $\mathcal{C}.$ There exist an exact sequence of group
$$ \begin{array}{ccccccccc} 0 & \longrightarrow & \prod_{v\in \mathbf{V}(\Gamma)} Aut(\tilde{\mathcal{C}}_v,(q_h)_{h\in\mathbf{H}(v)})& \longrightarrow Aut(\mathcal{C},p_1,\cdots,p_n) & \longrightarrow &Aut(\Gamma) \longrightarrow & 0 \end{array} $$
I need to show the sequence is exact and my problem is that if I take $(\tilde{ \sigma}_{v})_{v\in \mathbf{V}(\Gamma)} \in \prod_{v\in \mathbf{V}(\Gamma)} Aut(\tilde{\mathcal{C}}_v,(q_h)_{h\in \mathbf{H}(v)}$ how to send it to $\sigma \in Aut(\mathcal{C},p_1,\cdots,p_n)$ such that it has to be stable.