Let (X, d) and (Y, ρ) be compact metric spaces, let τ and η be Lipschitz involutions on X and Y , respectively, let A = Lip(X, d, τ ), and let B = Lip(Y, ρ, η), where Lip(X, d, τ ) = {f ∈ Lip(X, d) : f ◦ τ = ¯ f}. For each f ∈ A, σπ,A(f) denotes the peripheral spectrum of f. We prove that if S1, S2 : A → A and T1, T2 : A → B are surjective mappings that satisfy σπ,B(T1(f)T2(g)) = σπ,A(S1(f)S2(g)) for all f, g ∈ A, then there are κ1, κ2 ∈ Lip(Y, ρ, η) with κ1κ2 = 1Y and a Lipschitz homeomorphism φ from (Y, ρ) to (X, d) with τ ◦ φ = φ ◦ η on Y such that Tj(f) = κj · (Sj(f) ◦ φ) for all f ∈ A and j = 1, 2. Moreover, we show that the same result holds for surjective mappings S1, S2 : A → A and T1, T2 : A → B that satisfy σπ,B(T1(f)T2(g)) ∩ σπ,A(S1(f)S2(g)) ̸= ∅ for all f, g ∈ A.
