Let $G$ be a finite group and ${\rm\Gamma}$ a $G$-symmetric graph. Suppose that $G$ is imprimitive on $V({\rm\Gamma})$ with $B$ a block of imprimitivity and ${\mathcal{B}}:=\{B^{g};g\in G\}$ a system of imprimitivity of $G$ on $V({\rm\Gamma})$. Define ${\rm\Gamma}_{{\mathcal{B}}}$ to be the graph with vertex set ${\mathcal{B}}$ such that two blocks $B,C\in {\mathcal{B}}$ are adjacent if and only if there exists at least one edge of ${\rm\Gamma}$ joining a vertex in $B$ and a vertex in $C$. Xu and Zhou [‘Symmetric graphs with 2-arc-transitive quotients’, J. Aust. Math. Soc. 96 (2014), 275–288] obtained necessary conditions under which the graph ${\rm\Gamma}_{{\mathcal{B}}}$ is 2-arc-transitive. In this paper, we completely settle one of the cases defined by certain parameters connected to ${\rm\Gamma}$ and ${\mathcal{B}}$ and show that there is a unique graph corresponding to this case.

A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let p be a prime. It was shown by Folkman [J. Folkman, ‘Regular line-symmetric graphs’, J. Combin. Theory3 (1967), 215–232] that a regular edge-transitive graph of order 2p or 2p2 is necessarily vertex-transitive. In this paper an extension of his result in the case of cubic graphs is given. It is proved that every cubic edge-transitive graph of order 8p2 is vertex-transitive.

An automorphism group of a graph is said to be s-regular if it acts regularly on the set of s-arcs in the graph. A graph is s-regular if its full automorphism group is s-regular. For a connected cubic symmetric graph X of order 2pn for an odd prime p, we show that if p ≠ 5, 7 then every Sylow p-subgroup of the full automorphism group Aut(X) of X is normal, and if p ≠3 then every s-regular subgroup of Aut(X) having a normal Sylow p-subgroup contains an (s − 1)-regular subgroup for each 1 ≦ s ≦ 5. As an application, we show that every connected cubic symmetric graph of order 2pn is a Cayley graph if p > 5 and we classify the s-regular cubic graphs of order 2p2 for each 1≦ s≦ 5 and each prime p. as a continuation of the authors' classification of 1-regular cubic graphs of order 2p2. The same classification of those of order 2p is also done.