Semilinear automorphisms of reductive algebraic groups

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.