On Arc-Transitive Regular Covers of Cubic Edge-Primitive Graphs

We determine, up to isomorphism of the covering graphs, the connected arc-transitive regular covers of cubic edge-primitive graphs whose covering transformation group is cyclic or elementary abelian of order $p^2$, where $p$ is a prime. Combining the known classifications for the base graphs ${\mathrm{K}}_{3,3}$ and $\mathrm{D}{\mathrm{C}}_{14}$ with new arguments for $\mathrm{F}30\mathrm{A}$ and $\mathrm{F}102\mathrm{A}$ gives the full list in these two classes of covering groups. In the cyclic case, the covers of $\mathrm{F}30\mathrm{A}$ and $\mathrm{F}102\mathrm{A}$ are $\mathrm{F}90\mathrm{A}$ and $\mathrm{F}204\mathrm{A}$, respectively. In the elementary abelian case, neither $\mathrm{F}30\mathrm{A}$ nor $\mathrm{F}102\mathrm{A}$ admits an arc-transitive regular ${\mathbb{Z}}_p^2$-cover, so the base graph is ${\mathrm{K}}_{3,3}$ or $\mathrm{D}{\mathrm{C}}_{14}$.