  Let $ G $ be a connected reductive algebraic group over a field $ k $. We
study the group of semilinear automorphisms Aut($ G\to $Spec $k$) consisting of
algebraic automorphisms of $ G $ over automorphisms of $ k $. We focus on the
exact sequence $ 1\to $Aut $G\to $Aut ($ G\to $Spec $k$)$\to $Aut$_{G}(k)\to 1
$. When $G$ is quasi-split, we show that Aut$_{G}(k)$ is isomorphic to
Aut$_{\mathcal{R}(G)}(k)$, where $\mathcal{R}(G)$ denotes the scheme of based
root datum of $G$. Furthermore, the exact sequence $ 1\to $Aut $G\to $Aut ($
G\to $Spec $k$)$\to $Aut$_{G}(k)\to 1 $ splits if and only if the exact
sequence $ 1\to \text{Aut }\mathcal{R}(G) \to \text{Aut }(\mathcal{R}(G) \to
\text{Spec } k)\to \text{Aut}_{\mathcal{R}(G)}(k)\to 1 $ splits. As a
corollary, we get many examples of algebraic groups $ G $ over $ k $ whose
group of abstract automorphisms does not decompose as the semidirect product of
$ \text{Aut } G $ with $\text{Aut}_G(k) $. We also study the same questions for
inner forms of SL$_n $ over a local field.
