Lifts of Automorphisms of Symmetric Digraphs Associated With Cyclic Abelian Covers III

Let \(D\) be a connected symmetric digraph, \(\Gamma\) a group of automorphisms of \(D\), and \(A\) a finite abelian group. For a cyclic \(A\)-cover of \(D\), we consider a lift of \(\gamma \in \Gamma\), and the associated group automorphism of some subgroup of \(Aut \, A\). Furthermore, we give a characterization for any \(\gamma \in \Gamma\) to have a lift in terms of some matrix.