Real automorphisms of the "quaternionic" real group ${\rm SO}^*(4m)$ Let $m\ge 2$, and let $G={\rm SO}^*(4m)$ denote the "quaternionic" real form of the special orthogonal group ${\rm SO}(4m,\mathbb C)$ of type ${\sf D}_{2m}$.
Let $\tau\in{\rm Aut}_{\Bbb R}(G)$ be a real automorphism of $G$, that is,
 an automorphism defined over $\Bbb R$.
My Galois-cohomological calculations suggest that then $\tau$ is an inner automorphism.

Question.
  Is it true that, although $G$ does have complex outer automorphisms, it has no real outer automorphisms? 

Clarification: I regard $G={\rm SO}^*(4m)$  as an algebraic group over $\Bbb R$.  By a complex inner automorphism of $G$ I mean an element of the group ${\rm Inn}(G)(\Bbb C)$, where ${\rm Inn}(G)=G/Z(G)$. By a real inner automorphism of $G$ I mean an element of ${\rm Inn}(G)(\Bbb R)$.
 A: No, there are no real outer automorphisms of $G = SO^*(4m)$.  Suppose, for sake of contradiction, that one exists and call it $\phi$.  One of the half-spin representations $\rho \!: G \to GL(V)$ is real, and the composition $\rho \phi$ provides an irreducible representation defined over $\mathbb{R}$ of $G$ that is (by examining highest weights) the other half-spin representation.  But the other half-spin representation is quaternionic, a contradiction.  
(The fact that one half-spin rep of $SO^*(4m)$ is real and the other is quaternionic also played a role in this related MO question "The Tits classes of simply connected simple real groups".)
Generalizing, we can replace $k = \mathbb{R}$ with any field and $G$ with a semisimple $k$-group.  The general statement is that the Tits algebras of $G$ provide an obstruction to the existence of outer $k$-automorphisms of $G$.  See Theorem 11 in my paper Outer automorphisms of algebraic groups and determining groups by their maximal tori (Michigan Mathematical Journal 61 #2 (2012), 227-237).  The specific example of groups of type $^1D_n$ as in this question is at the top of page 233.
