On $G$-vertex-transitive covers of complete graphs having at most two $G$-orbits on the arc set

We investigate abelian (in the sense of Godsil and Hensel) distance-regular covers of complete graphs with the following property: there is a vertex-transitive group of automorphisms of the cover which possesses at most two orbits in the induced action on its arc set. We focus on covers whose parameters belong to some known infinite series of feasible parameters. We also complete the classification of arc-transitive covers with a non-solvable automorphism group and show that the automorphism group of any unknown edge-transitive cover induces a one-dimensional affine permutation group on the set of its antipodal classes.