Determining the vertex stabilizers of 4-valent half-arc-transitive graphs

We say that a group is a 4-HAT-stabilizer if it is the vertex stabilizer of some connected 4-valent half-arc-transitive graph. In 2001, Marušič and Nedela proved that every 4-HAT-stabilizer must be a concentric group. However, over the past two decades, only a very small proportion of concentric groups have been shown to be 4-HAT-stabilizers. This paper develops a theory that provides a general framework for determining whether a concentric group is a 4-HAT-stabilizer. With this approach, we significantly extend the known list of 4-HAT-stabilizers. As a corollary, we confirm that $H_7\times C_2^{m-7}$ are 4-HAT-stabilizers for $m\ge 7$, achieving the goal of a conjecture posed by Spiga and Xia.