The symmetries of McCullough–Miller space

We prove that if $W$ is the free product of at least four groups of order $2$, then the automorphism group of the McCullough-Miller space corresponding to $W$ is isomorphic to group of outer automorphisms of $W$. We also prove that, for each integer $n \geq 3$, the automorphism group of the hypertree complex of rank $n$ is isomorphic to the symmetric group of rank $n$.