Let $X$ be a normal affine variety of dimension at least two over $\mathbb{C}$ and let $U\subset X$ be a dense open. Assume that $\mathrm{codim}(X\setminus U) \geq 2$.

I think Hartog's lemma implies that every automorphism of $U$ extends to an automorphism of $X$. Is that true?

Do we have a surjective homomorphism $\mathrm{Aut}(X) \to \mathrm{Aut}(U)$?