Let $G={SL}_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$. Let $H\subset G$ be a finite subgroup. Set $X=G/H$ be the corresponding homogeneous space, it is a complex variety. Let $\tau\in {\rm Aut}({\mathbb{C}})$ be an automorphism of the field of complex numbers, not necessarily continuous. Consider the conjugate variety $\tau X$.

Question 1. Is it always true that $\tau X\simeq X$ as algebraic varieties over ${\mathbb{C}}$?

Question 2. Is it always true that that the $(\tau X)({\mathbb{C}})\simeq X({\mathbb{C}})$ as $C^\infty$-manifolds?

(I expect the answer NO, at least to Question 1.}