1) The image $\Gamma_{\rho_{X}}$ of $Aut(X)$ in $Aut(\Lambda_{X},\rho_{X})$ is a subgroup of $\Gamma_{v_{X}}$ where $v_{X}:=val_{K}\circ\rho_{X}:\Lambda_{X}\to{{\bf R}}$.

2) The homomorphism $Aut(X)\to Aut_{{{\bf Z}}PL,v_{X}}(S^{2})$ is conjugate to the restriction to $\Gamma_{\rho_{X}}$ of the homomorphism $\phi_{v_{X}}$ defined in the previous subsection.