Classification of equivariant quasi-local automorphisms on quantum chains

We classify automorphisms on quantum chains, allowing both spin and fermionic degrees of freedom, that are moreover equivariant with respect to a local symmetry action of a finite symmetry group $G$. The classification is up to equivalence through strongly continuous deformation and stacking with decoupled auxiliary automorphisms. We find that the equivalence classes are uniquely labeled by an index taking values in $\mathbb{Q} \cup \sqrt{2} \mathbb{Q} \times \mathrm{Hom}(G, \mathbb{Z}_2) \times H^2(G, U(1))$. We discuss te relation of this index to the index of one-dimensional symmetry protected topological phases on spin chains, which takes values in $H^2(G, U(1))$.