∀ {A : Type u_1} {B : Type u_2} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] {G : Type u_3}
  [inst_3 : Group G] [inst_4 : MulSemiringAction G B] [inst_5 : SMulCommClass G A B] [inst_6 : TopologicalSpace G]
  [CompactSpace G] [IsTopologicalGroup G] (P : Ideal A) (Q : Ideal B) [Q.IsPrime] [inst_10 : Q.LiesOver P]
  [Algebra.IsInvariant A B G] (σ : (B ⧸ Q) ≃ₐ[A ⧸ P] B ⧸ Q) (N : OpenNormalSubgroup G),
  Nonempty ((Ideal.Quotient.stabilizerHomSurjectiveAuxFunctor P Q σ).obj N)