19.98 (2018)

A connected graph $\Sigma$ is a symmetrical extension of a graph $\Gamma$ by a graph $\Delta$ if there exist a vertex-transitive group $G$ of automorphisms of $\Sigma$ and an imprimitivity system $\sigma$ of $G$ on the set of vertices of $\Sigma$ such that the quotient graph $\Sigma/\sigma$ is isomorphic to $\Gamma$ and blocks of $\sigma$ generate in $\Sigma$ subgraphs isomorphic to $\Delta$.
$\qquad$ (a) Let $\Gamma$ be a locally finite Cayley graph of a finitely presented group, and $\Delta$ a finite graph. Are there only finitely many symmetrical extensions of $\Gamma$ by $\Delta$?
$\qquad$ (b) Let $\Gamma$ be a locally finite graph which has the property of $k$-contractibility for some positive integer $k$ (see the definition in (V. I. Trofimov, Proc. Steklov Inst. Math., 279, suppl. 1 (2012), 107–112); note that any $\Gamma$ from (a) is such a graph) and let $\Delta$ be a finite graph. Are there only finitely many symmetrical extensions of $\Gamma$ by $\Delta$?