Proposition 1

The group morphism $\beta_{a_0}: G_{a_0}\rightarrow G'_{\beta(a_0)}$ is surjective, and its kernel is the smallest subgroup of $G_{a_0}$ containing the conjugacy classes $c(\mathbf{z})$ for $\mathbf{z}\in \Omega (\widetilde{\Gamma})$.

Let K be the smallest distinguished subgroupoid of G whose set of arrows contains the image of $h$. The morphism $G_{a_0}\rightarrow G'_{\beta(a_0)}$ is surjective and its kernel is equal to $K_{a_0}($II, p. 170, prop. 2). The proposition then follows from prop. 8 of II, p. 176.