The property of $G$ you are looking at is called 2-closure, i.e., you ask for a classification of 2-closed permutation groups. See the paper by Liebeck, Praeger, Saxl, On the 2-closures of finite permutation groups. J. London Math. Soc. (2) 37 (1988), no. 2, 241–252, where this question is investigated for a particular class of primitive permutation groups.
More precisely, this is not 100% equivalent to your question (it would be, if you allowed graphs have coloured arcs), as there exist 2-closed permutation groups $G$ such that for each orbital the automorphism group of the underlying (di)graph is strictly bigger than $G$.
In general such a classification is not known, and the problem is hopelessly hard, I think.