Some remarks on symmetrical extensions of graphs

It is proved that results from a previous paper of the author on symmetrical $2$-extensions of graphs can be extended to symmetrical $p$-extensions of graphs for any prime $p$. In particular, it is proved that for any prime $p$ there are only finitely many symmetrical $p$-extensions of a locally finite graph which has an abelian subgroup of finite index in its automorphism group. Some refinements of these results are also proved. In addition, it is considered a question on the possibility to represent symmetrical extensions of a $d$-dimension grid (and similar graphs) in the $d$-dimension affine Euclidean space in such a way that a corresponding vertex-transitive group of automorphisms of the extension is induced by some crystallographic group of the space.