Double cosets for $\text{SU}(2)\times \mathrm{.}\mathrm{.}\mathrm{.}\times \text{SU}(2)$ and outer automorphisms of free groups

Consider the space of double cosets of the product of $n$ copies of $\text{SU}(2)$ with respect to the diagonal subgroup. We get a parametrization of this space, the radial part of the Haar measure, and explicit formulas for the actions of the group of outer automorphisms of the free group $F_{n-1}$ and of the braid group of $n-1$ strings.