For an odd prime
$p$
, a
$p$

transposition group
is a group generated by a set of involutions such that the product of any two has order 2 or
$p$
. We first classify a family of
$(G,2)$
geodesic transitive Cayley graphs
${\rm\Gamma}:=\text{Cay}(T,S)$
where
$S$
is a set of involutions and
$T:\text{Inn}(T)\leq G\leq T:\text{Aut}(T,S)$
. In this case,
$T$
is either an elementary
... [Show full abstract] abelian 2group or a
$p$
transposition group. Then under the further assumption that
$G$
acts quasiprimitively on the vertex set of
${\rm\Gamma}$
, we prove that: (1) if
${\rm\Gamma}$
is not
$(G,2)$
arc transitive, then this quasiprimitive action is the holomorph affine type; (2) if
$T$
is a
$p$
transposition group and
$S$
is a conjugacy class, then
$p=3$
and
${\rm\Gamma}$
is
$(G,2)$
arc transitive.