Constructing infinitely many half-arc-transitive covers of tetravalent graphs

We prove that, given a finite graph Σ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of Σ. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order 28 of finite connected tetravalent half-arc-transitive graphs. This sheds some new light on the longstanding problem of classifying the vertex stabilizers of finite tetravalent half-arc-transitive graphs.