Symmetric graphs with 2-arc transitive quotients

A graph $\text{Ga}$ is $G$-symmetric if $\text{Ga}$ admits $G$ as a group of automorphisms acting transitively on the set of vertices and the set of arcs of $\text{Ga}$, where an arc is an ordered pair of adjacent vertices. In the case when $G$ is imprimitive on $V(\text{Ga})$, namely when $V(\text{Ga})$ admits a nontrivial $G$-invariant partition $\text{BB}$, the quotient graph ${\text{Ga}}_{\text{BB}}$ of $\text{Ga}$ with respect to $\text{BB}$ is always $G$-symmetric and sometimes even $(G,2)$-arc transitive. (A $G$-symmetric graph is $(G,2)$-arc transitive if $G$ is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for ${\text{Ga}}_{\text{BB}}$ to be $(G,2)$-arc transitive (regardless of whether $\text{Ga}$ is $(G,2)$-arc transitive) in the case when $v-k$ is an odd prime $p$, where $v$ is the block size of $\text{BB}$ and $k$ is the number of vertices in a block having neighbours in a fixed adjacent block. These conditions are given in terms of $v,k$ and two other parameters with respect to $(\text{Ga},\text{BB})$ together with a certain 2-point transitive block design induced by $(\text{Ga},\text{BB})$. We prove further that if $p=3$ or $5$ then these necessary conditions are essentially sufficient for ${\text{Ga}}_{\text{BB}}$ to be $(G,2)$-arc transitive.