Suppose $G$ is a connected reductive algebraic group over an arbitrary field $K$; let $Z$ be the center of $G$. The inner automorphisms of $G$ are given by $\operatorname{Inn}(G) = G / Z = G^{\operatorname{ad}}$. Set $\operatorname{Out}(G)$ to be the quotient $\operatorname{Aut}(G) / G^{\operatorname{ad}}.$ The forms of $G$ are parameterized by $\operatorname{H}^1(K, \operatorname{Aut}(G))$, and the inner forms are those in the image of

$$\operatorname{H}^1(K,G^{\operatorname{ad}}) \rightarrow \operatorname{H}^1(K,\operatorname{Aut}(G))$$
So we can recast the classification of inner forms of $G$ to:

What conditions can we put on $G$ to guarantee that the map $$\operatorname{H}^0(K,\operatorname{Aut}(G)) \rightarrow \operatorname{H}^0(K,\operatorname{Out}(G))$$ is surjective?

I'm primarily interested in the connected reductive case here, but I would be curious about the more general case as well.

On a related note, I've frequently seen the claim that for quasisimple $G$, the group $\operatorname{Out}(G)$ is given by the automorphism group of the Dynkin diagram of $G$. This holds for some reductive $G$ (such as $\operatorname{GL}_n$) and not others (most nontrivial tori will have a nontrivial outer automorphism group and a trivial dynkin diagram).

How can we extend the description of $\operatorname{Out}(G)$ from the case of quasi-simple $G$ to connected reductive $G$?