  The question of existence of outer automorphisms of a simple algebraic group
$G$ arises naturally both when working with the Galois cohomology of $G$ and as
an example of the algebro-geometric problem of determining which connected
components of the automorphism group of $G$ have rational points. The existence
question remains open only for four types of groups, and we settle one of the
remaining cases, type $^3D_4$. The key to the proof is a Skolem-Noether theorem
for cubic etale subalgebras of Albert algebras which is of independent
interest. Necessary and sufficient conditions for a simply connected group of
outer type $A$ to admit outer automorphisms of order 2 are also given.
