Geodesic
The structure of automorphism groups of Cayley graphs and maps.
Journal of Algebraic Combinatorics, Tome 12 (2000) no. 1, pp. 73-84.

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Summary: The automorphism groups $Aut( C( G, X))$ and $Aut( CM( G, X, p))$ of a Cayley graph $C( G, X)$ and a Cayley map $CM( G, X, p)$ both contain an isomorphic copy of the underlying group $G$ acting via left translations. In our paper, we show that both automorphism groups are rotary extensions of the group $G$ by the stabilizer subgroup of the vertex $1 _{ G }$. We use this description to derive necessary and sufficient conditions to be satisfied by a finite group in order to be the (full) automorphism group of a Cayley graph or map and classify all the finite groups that can be represented as the (full) automorphism group of some Cayley graph or map.
Keywords: Cayley graph, Cayley map, automorphism group 
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Jajcay, Robert. The structure of automorphism groups of Cayley graphs and maps.. Journal of Algebraic Combinatorics, Tome 12 (2000) no. 1, pp. 73-84. http://geodesic.mathdoc.fr/item/JAC_2000__12_1_a2/