  For a classical group $G$ of type $\mathsf D_n$ over a field $k$ of
characteristic different from $2$, we show the existence of a finitely
generated regular extension $R$ of $k$ such that $G$ admits outer automorphisms
over $R$. Using this result and a construction of groups of type $\mathsf A$
from groups of type $\mathsf D$, we construct new examples of groups of type
$^2\mathsf A_n$ with $n\equiv 3\bmod 4$ and the first examples of type
$^2\mathsf A_n$ with $n\equiv 1\bmod 4$ $(n\geq 5)$ that are not $R$-trivial,
hence not rational (nor stably rational).
