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Let $W$ be a finite Weyl group and $\sg$ be a non-trivial graph automorphismof $W$. We show a remarkable relation between the $\sg$-twisted involutionmodule for $W$ and the Frobenius--Schur indicators of the unipotent charactersof a corresponding twisted finite group of Lie type. This extends earlierresults of Lusztig-Vogan for the untwisted case and then allows us to state ageneral result valid for any finite group of Lie type. Inspired by recent workof Marberg, we also formally define Frobenius--Schur indicators for "unipotentcharacters" of twisted dihedral groups.

Frobenius--Schur indicators of unipotent characters and the twisted involution module