This paper investigates the automorphism group of a connected and undirected G-symmetric graph Γ where G is an almost simple group with socle T. First we prove that, for an arbitrary subgroup M of containing G, either T is normal in M or T is a subgroup of the alternating group A_k of degree . Then we describe the structure of the full automorphism group of G-locally primitive graphs of valency d, where d≤20 or is a prime. Finally, as one of the applications of our results, we determine the structure of the automorphism group for cubic symmetric graph Γ admitting a finite almost simple group.