EDITED 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 $\tau X$.

We have $\tau X=\tau G/\tau H$. We take the standard $\mathbb{Q}$-form ${SL}_{n,\mathbb{Q}}$ of $G$, then we identify $\tau G$ with $G$, and we obtain $\tau X= G/\tau(H)$.

Question 1. Is it always true that $\tau X\simeq X$ as homogeneous spaces of $G={SL}_{n,\mathbb{C}}$?

Question 2. Is it always true that $\tau X\simeq X$ as algebraic varieties over ${\mathbb{C}}$?

Question 3. Is it always true that that the $(\tau X)({\mathbb{C}})\simeq X({\mathbb{C}})$ as $C^\infty$-manifolds?

I expect the answer NO to Question 1.
Let us try to construct a counter-example. We wish to construct a finite subgroup $H\subset G$ such that $G/\tau(H)\not\simeq G/H$, i.e., $\tau(H)$ is not conjugate to $H$ in $SL(n,\mathbb{C})$. We come to the following question:

Question 4. Let $\rho\colon H\to SL(n,\mathbb{C})$ be an $n$-dimensional complex representation of a finite group $H$. Is it possible that $\tau(\rho(H))$ is not conjugate to $\rho(H)$ in $SL(n,\mathbb{C})$ for some $\tau\in {\rm Aut}({\mathbb{C}})$?

In the last question by $\tau(\rho(H))$ we mean $\{ \tau(\rho(h))\, |\ h\in H\}$, where $\tau(\rho(h))$ is the matrix with matrix elements $\tau(\rho(h)_{i,j})$.

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$EDITED 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 $\tau\in \Aut({\mathbb{C}})$ be an automorphism of the field of complex numbers, not necessarily continuous. Consider the conjugate variety $\tau X$.

We have $\tau X=\tau G/\tau H$. We take the standard $\mathbb{Q}$-form $\SL_{n,\mathbb{Q}}$ of $G$, then we identify $\tau G$ with $G$, and we obtain $\tau X= G/\tau(H)$.

Question 1. Is it always true that $\tau X\simeq X$ as homogeneous spaces of $G=\SL_{n,\mathbb{C}}$?

Question 2. Is it always true that $\tau X\simeq X$ as algebraic varieties over ${\mathbb{C}}$?

Question 3. Is it always true that that the $(\tau X)({\mathbb{C}})\simeq X({\mathbb{C}})$ as $C^\infty$-manifolds?

I expect the answer NO to Question 1.
Let us try to construct a counter-example. We wish to construct a finite subgroup $H\subset G$ such that $G/\tau(H)\not\simeq G/H$, i.e., $\tau(H)$ is not conjugate to $H$ in $\SL(n,\mathbb{C})$. We come to the following question:

Question 4. Let $\rho\colon H\to \SL(n,\mathbb{C})$ be an $n$-dimensional complex representation of a finite group $H$. Is it possible that $\tau(\rho(H))$ is not conjugate to $\rho(H)$ in $\SL(n,\mathbb{C})$ for some $\tau\in \Aut({\mathbb{C}})$?

In the last question by $\tau(\rho(H))$ we mean $\{ \tau(\rho(h))\, |\ h\in H\}$, where $\tau(\rho(h))$ is the matrix with matrix elements $\tau(\rho(h)_{i,j})$.

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.)

EDITED 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 $\tau X$.

We have $\tau X=\tau G/\tau H$. We take the standard $\mathbb{Q}$-form ${SL}_{n,\mathbb{Q}}$ of $G$, then we identify $\tau G$ with $G$, and we obtain $\tau X= G/\tau(H)$.

Question 1. Is it always true that $\tau X\simeq X$ as homogeneous spaces of $G={SL}_{n,\mathbb{C}}$?

Question 2. Is it always true that $\tau X\simeq X$ as algebraic varieties over ${\mathbb{C}}$?

Question 3. Is it always true that that the $(\tau X)({\mathbb{C}})\simeq X({\mathbb{C}})$ as $C^\infty$-manifolds?

I expect the answer NO to Question 1.
Let us try to construct a counter-example. We wish to construct a finite subgroup $H\subset G$ such that $G/\tau(H)\not\simeq G/H$, i.e., $\tau(H)$ is not conjugate to $H$ in $SL(n,\mathbb{C})$. We come to the following question:

Question 4. Let $\rho\colon H\to SL(n,\mathbb{C})$ be an $n$-dimensional complex representation of a finite group $H$. Is it possible that $\tau(\rho(H))$ is not conjugate to $\rho(H)$ in $SL(n,\mathbb{C})$ for some $\tau\in {\rm Aut}({\mathbb{C}})$?

In the last question by $\tau(\rho(H))$ we mean $\{ \tau(\rho(h))\, |\ h\in H\}$, where $\tau(\rho(h))$ is the matrix with matrix elements $\tau(\rho(h)_{i,j})$.

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.}

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.)