Automorphisms of Coxeter groups of type $K_n$

A Coxeter system $(W,S)$ is said to be of type $K_n$ if the associated Coxeter graph $\Gamma_S$ is complete on $n$ vertices and has only odd edge labels. If $W$ satisfies either of: (1) $n=3$; (2) $W$ is rigid; then the automorphism group of $W$ is generated by the inner automorphisms of $W$ and any automorphisms induced by $\Gamma_S$. Indeed, $\operatorname{Aut}(W)$ is the semidirect product of $\operatorname{Inn}(W)$ and the group of diagram automorphisms, and furthermore $W$ is strongly rigid. We also show that if $W$ is a Coxeter group of type $K_n$ then $W$ has exactly one conjugacy class of involutions and hence $\operatorname{Aut}(W)=\operatorname{Spec}(W)$.