The Cayley graphs of $\mathbb Z^d$ and the limits of vertex-primitive graphs of $HA$-type

We study the limits of the finite graphs that admit some vertex-primitive group of automorphisms with a regular abelian normal subgroup. It was shown in [1] that these limits are Cayley graphs of the groups $\mathbb Z^d$. In this article we prove that for each $d>1$ the set of Cayley graphs of $\mathbb Z^d$ presenting the limits of finite graphs with vertex-primitive and edge-transitive groups of automorphisms is countable (in fact, we explicitly give countable subsets of these limit graphs). In addition, for $d<4$ we list all Cayley graphs of $\mathbb Z^d$ that are limits of minimal vertex-primitive graphs. The proofs rely on a connection of the automorphism groups of Cayley graphs of $\mathbb Z^d$ with crystallographic groups.