I currently study the strata of the moduli stack of algebraic curves and have understood, that for a stable curve with markings $p_1, \dots, p_n$ there is a morphism of groups $Aut(C, p_1, \dots, p_n) \to Aut(\Gamma)$, where $\Gamma$ is the associated stable graph. I also see, that because of automorphisms of the connected components of the normalisation that are not "seen" by the graph, in general this morphism is not injective.
But is it surjective? In all the examples I have considered I think it is, but I do not see a general argument to show that, so I am thankful for any hints.