Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$. [Edit: Assume $S$ contains only algebraic numbers.]

Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

That is, does there exist a curve $X$, a finite etale morphism $X\to \mathbb{A}^1-S$ and a finite etale morphism $X\to \mathbb{A}^1-\{0,1\}$?

For which $S$ is the answer positive?

We work over the field of complex numbers. (But remarks in characteristic $p$ are very welcome.)

Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$. [Edit: Assume $S$ contains only algebraic numbers.]

Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

That is, does there exist a curve $X$, a finite etale morphism $X\to \mathbb{A}^1-S$ and a finite etale morphism $X\to \mathbb{A}^1-\{0,1\}$?

For which $S$ is the answer positive?

Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$.

Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

That is, does there exist a curve $X$, a finite etale morphism $X\to \mathbb{A}^1-S$ and a finite etale morphism $X\to \mathbb{A}^1-\{0,1\}$?

For which $S$ is the answer positive?

Let $S$ be a finite set of points in $\mathbb{A}^1$ containing $0$ and $1$. [Edit: Assume $S$ contains only algebraic numbers.]

Do $\mathbb{A}^1-S$ and $\mathbb{A}^1-\{0,1\}$ have a finite etale cover in common?

That is, does there exist a curve $X$, a finite etale morphism $X\to \mathbb{A}^1-S$ and a finite etale morphism $X\to \mathbb{A}^1-\{0,1\}$?

For which $S$ is the answer positive?