  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 $H^2(G, U(1))$.
