  The so-called Tits class, associated to an adjoint absolutely almost simple
algebraic group, provides a cohomological obstruction for this group to admit
an outer automorphism. If the group has inner type, this obstruction is the
only one. In this paper, we prove this is not the case for classical groups of
outer type, except for groups of type $^2\mathsf{A}_n$ with $n$ even, or $n=5$.
More precisely, we prove a descent theorem for exponent $2$ and degree $6$
algebras with unitary involution, which shows that their automorphism groups
have outer automorphisms. In all other relevant classical types, namely
$^2\mathsf{A}_n$ with $n$ odd, $n\geq3$ and $^2\mathsf{D}_n$, we provide
explicit examples where the Tits class obstruction vanishes, and yet the group
does not have outer automorphism. As a crucial tool, we use "generic" sums of
algebras with involution.
