Let $(X,d)$ and $(Y,rho)$ be compact metric spaces, $tau$ and $eta$ be Lipschitz involutions on $ X$ and $Y$, respectively, $mathcal{A}=Lip(X,d,tau)$ and $mathcal{B}=Lip(Y,rho,eta)$, where $Lip(X,d,tau)=lbrace fin Lip(X,d):fcirctau=bar{f}rbrace $. For each $fin mathcal{A}$, $sigma_{pi,mathcal{A}}(f)$ denotes the peripheral spectrum of $f$. We prove that if $S_{1},S_{2}:mathcal{A}rightarrow mathcal{A}$ and $T_{1},T_{2}:mathcal{A}rightarrow mathcal{B}$ are surjective mappings that satisfy $sigma_{pi,mathcal{B}}(T_{1}(f)T_{2}(g))=sigma_{pi,mathcal{A}}(S_{1}(f)S_{2}(g))$ for all $f,gin mathcal{A}$, then there are $kappa_{1},kappa_{2}in Lip(Y,rho,eta)$ with $kappa_{1}kappa_{2}=1_{Y}$ and a Lipschitz homeomorphism $varphi$ from $(Y,rho)$ to $(X,d)$ with $tau circvarphi=varphi circ eta$ on $Y$ such that $T_{j}(f)=kappa_{j}cdot(S_{j}(f)circvarphi)$ for all $fin mathcal{A}$ and $j=1,2$. Moreover, we show that the same result holds for surjective mappings $S_{1},S_{2}:mathcal{A}rightarrow mathcal{A}$ and $T_{1},T_{2}:mathcal{A}rightarrow mathcal{B}$ that satisfy $sigma_{pi,mathcal{B}}(T_{1}(f)T_{2}(g))capsigma_{pi,mathcal{A}}(S_{1}(f)S_{2}(g))neqemptyset$ for all $f,gin mathcal{A}$.