Two allowable subgroups M_1/R^{\ast} and M_2/R^{\ast} are called equivalent if the quotients F/M_1\simeq F/M_2, that are the corresponding immediate descendants of G, are isomorphic.
Since \alpha^\ast(M/R^{\ast})\cdot P_c(F/R^{\ast})=\alpha^\ast\lbrack M/R^{\ast}\cdot P_c(F/R^{\ast})\rbrack=\alpha^\ast(R/R^\ast)=R/R^\ast, each extended automorphism \alpha^\ast\in\mathrm{Aut}(G^\ast) induces a permutation \alpha^\prime of the allowable subgroups M/R^{\ast}\le R/R^{\ast}.