Definition
Let \(X\) be an Artin stack over \(S\) with \(G\)-action. For any scheme \(A\) and every \(A\)-valued point \(x\) of \(X\), the \(G\)-stablizer (or stabilizer of the \(G\)-action) at \(x\) is an fppf sheaf of groups \(\St^G_X(s)\) defined as the cokernel of the homomorphism \(\underline{\Aut}_X(x)\hookrightarrow\underline{\Aut}_{\mathcal X}(x)\), where \(\mathcal X := [X/G]\). Thus we have a short exact sequence \[1 \to \underline{\Aut}_X(x)\to \underline{\Aut}_{\mathcal X}(x)\to \St^G_X(x)\to 1\] of sheaves of groups over \(A\). Note that \(\St^G_X(s)\) can be regarded as a subgroup of \(G_A\), since it is the image of \(\alpha_A:\underline{\Aut}_{\mathcal X}(x)\to G_A\).