  We give a cohomological criterion for existence of outer automorphisms of a
semisimple algebraic group over an arbitrary field. This criterion is then
applied to the special case of groups of type D_2n over a global field, which
completes some of the main results from the paper "Weakly commensurable
arithmetic groups and isospectral locally symmetric spaces" (Pub. Math. IHES,
2009) by Prasad and Rapinchuk and gives a new proof of a result from another
paper by the same authors.
