Suppose $S_6$ is the symmetric group on six letters and let $X$ denote the conjugacy class containing $(12)(34)$. Define a graph $\Gamma$ with vertex set $X$ and edges precisely the 2-element subsets of $X$ which commute as elements of $S_6$. I would like to know the automorphism group of the graph $\Gamma$.

P.S. These graphs are known in scientific texts as 'commuting graphs'.