Let $A$ be a quaternion algebra over a field $K$ of characteristic zero. By the famous Skolem-Noether theorem, every $K$-algebra automorphism $\varphi \colon A \to A$ is of the form $$ \varphi(x) = gxg^{-1}, $$ for some invertible element $g \in A^\times$.
Question: Does it follow that also every group automorphism of $A^1 := \{ x \in A: \text{nrd}(x) = 1\}$ or of $A^\times$ is given by conjugation with some element of $A^\times$?
For example, if you take $A = M_2(\mathbb{R})$, then this is true because it is known that every group automorphism of $\text{SL}_2(\mathbb{R})$ or $\text{GL}_2(\mathbb{R})$ is given by conjugation with some element of $\text{GL}_2(\mathbb{R})$. However, most proofs available of this fact use very different methods and not the Skolem-Noether theorem as far as I know. So is it just a coincidence in this case?