  Let $S$ be a set of transpositions such that the girth of the transposition
graph of $S$ is at least 5. It is shown that the automorphism group of the
Cayley graph of the permutation group $H$ generated by $S$ is the semidirect
product $R(H) \rtimes \Aut(H,S)$, where $R(H)$ is the right regular
representation of $H$ and $\Aut(H,S)$ is the set of automorphisms of $H$ that
fixes $S$ setwise. Furthermore, if the connected components of the
transposition graph of $S$ are isomorphic to each other, then $\Aut(H,S)$ is
isomorphic to the automorphism group of the line graph of the transposition
graph of $S$. This result is a common generalization of previous results by
Feng, Ganesan, Harary, Mirafzal, and Zhang and Huang. As another special case,
we obtain the automorphism group of the extended cube graph that was proposed
as a topology for interconnection networks.
