Outer automorphisms of adjoint groups of type $\mathsf {D}$ and nonrational adjoint groups of outer type $\mathsf {A}$

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 of $k$ over which $G$ admits outer automorphisms. 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).