On automorphism groups of Cayley graphs

Let XG,H denote the Cayley graph of a finite group G with respect to a subset H. It is well-known that its automorphism group A(XG,H) must contain the regular subgroup LG corresponding to the set of left multiplications by elements of G. This paper is concerned with minimizing the index [A(XG,H):LG] for given G, in particular when this index is always greater than 1. If G is abelian but not one of seven exceptional groups, then a Cayley graph of G exists for which this index is at most 2. Nearly complete results for the generalized dicyclic groups are also obtained.