4-fold regular coverings of wedge of two circles

Question

Problem 7 (p.80) of Algebraic Topology book by Tammo Tom asks to classify all 4-fold regular coverings of a wedge of two circles. I am aware that the required coverings correspond to normal subgroups of a free group on 2 generators and that there is a way of finding these subgroups e.g. see here. While this method is easy to carry out for two or three-fold coverings, it is very tedious computationally in the four-fold case. However we just need the regular coverings, so i am wondering if there is a quicker, elegant and conceptual way to solve this problem. Will greatly appreciate any help. Thanks

Answer 1

Well, you can draw each possible regular cover directly: it'll be a graph with four vertices and 8 edges (some of which may be loops). With graphs this small, it's pretty easy to detect isomorphic ones.

You know that the automorphisms of the graph must be transitive on the four vertices. Perhaps there's an automorphism that sends 1->2->3->4; under that automorphism, each edge that meets vertex 1 is translated to one meeting vertex 2, etc....and pretty soon you can write down all possibilities for the graph.

By considering possible shuffles of the 4 verts, and how these shuffles must transform edges, I suspect you can come up with a full enumeration in not too long.