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.)