Outer automorphisms of adjoint groups of type D and non-rational adjoint groups of outer type A

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).