Automorphismes du groupe des automorphismes d'un groupe de Coxeter universel

Using the Guirardel-Levitt outer space of a free product, we prove that the outer automorphism group of the outer automorphism group of the universal Coxeter group of rank $n\ge 5$ is trivial, and that it is a cyclic group of order 2 if $n=4$. In addition we prove that the outer automorphism group of the automorphism group of the universal Coxeter group of rank $n\ge 4$ is trivial.