A graph
$\Gamma $
is
$G$
-symmetric if
$\Gamma $
admits
$G$
as a group of automorphisms acting transitively on the set of vertices and the set of arcs of
$\Gamma $
, where an arc is an ordered pair of adjacent vertices. In the case when
$G$
is imprimitive on
$V(\Gamma )$
, namely when
$V(\Gamma )$
admits a nontrivial
$G$
-invariant partition
${\mathcal{B}}$
, the quotient graph
$\Gamma _{\mathcal{B}}$
of
$\Gamma $
with respect to
${\mathcal{B}}$
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
$\Gamma _{{\mathcal{B}}}$
to be
$(G, 2)$
-arc transitive (regardless of whether
$\Gamma $
is
$(G, 2)$
-arc transitive) in the case when
$v-k$
is an odd prime
$p$
, where
$v$
is the block size of
${\mathcal{B}}$
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
$(\Gamma , {\mathcal{B}})$
together with a certain 2-point transitive block design induced by
$(\Gamma , {\mathcal{B}})$
. We prove further that if
$p=3$
or
$5$
then these necessary conditions are essentially sufficient for
$\Gamma _{{\mathcal{B}}}$
to be
$(G, 2)$
-arc transitive.