Suppose $S_6$ is symmetric group on six letters and consider subset $X$ is conjugacy class containing $(12)(34)$. Define graph $\Gamma$ such that its vertices is $X$ and between two vertices has an edge if they commute as two elements of $S_6$. I would like to know the automorphism group structure of graph $\Gamma$.

P.S. These graphs has known in scientific texts as commuting graph.