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

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