TITLE: Can every permutation group be realized as the automorphism group of a graph (acting on a subset of the vertices)?
QUESTION [6 upvotes]: By Frucht's theorem, every finite group can be realized as the automorphism group of a finite undirected graph. Because a permutation group is a finite group, it is clear that every permutation group be realized as the automorphism group of a graph. However, a permutation group acts on its underlying set, and I also want to realize this group action on a subset of the vertices of the graph.
An equivalent description of what I want is that I'm given a set of vertices and a permutation group acting on those vertices, and I want to construct a graph around those vertices such that the automorphism group of the graph realizes the given permutation group on the given set of vertices.

REPLY [4 votes]: FYI, this result exists in the literature. The two citations I know of are: 

L. Babai. Representation of permutation groups by graphs. In Combinatorial theory and its applications, I (Proc. Colloq., Balatonfured, 1969), pages 55–80. North- Holland, Amsterdam, 1970. 

and 

I. Z. Bouwer. Section graphs for finite permutation groups. J. Combinatorial Theory, 6:378–386, 1969.