  A graph is edge-transitive if its automorphism group acts transitively on the
edge set. In this paper, we investigate the automorphism groups of
edge-transitive graphs of odd order and twice prime valency. Let $\Gamma$ be a
connected graph of odd order and twice prime valency, and let $G$ be a subgroup
of the automorphism group of $\Ga$. In the case where $G$ acts transitively on
the edges and quasiprimitively on the vertices of $\Ga$, we prove that either
$G$ is almost simple or $G$ is a primitive group of affine type. If further $G$
is an almost simple primitive group then, with two exceptions, the socle of $G$
acts transitively on the edges of $\Gamma$.
