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? 
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**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 ${\rm Inn}(G)(\Bbb R)$.



Real automorphisms of the "quaternionic" real group ${\rm SO}^*(4m)$
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](https://mathoverflow.net/questions/291592/the-tits-classes-of-simply-connected-simple-real-groups/303176#303176)".)


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.


