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The quasi-split forms of a split reductive group $G$ over a field $k$ are classified by the elements of the first Galois cohomology group of $k$ with values in $Out(G)$ (see Theorem 23.51 of Milne's book Algebraic Groups). So no outer automorphisms means no nonsplit quasi-split forms. When $G$ is semisimple, the outer automorphisms correspond to graph automorphisms of the dynkin diagram.