I want to find some references for the following theorem. However, I do not know the exact statement of the theorem. So if there is anyone who can give me some references for this, it would be very helpful for me. Let me state the theorem to the extent that I know.

**Theorem(?).** Let $K$ be a field of characteristic $0$ and $\bar{K}$ be an algebraic closure of $K$ (if you want, you may assume that $K$ is a number field). Let $X$ be a geometrically irreducible smooth variety and $\bar{X}:=X\times_K\bar{K}$. Let $\tilde{\bar{X}}$ be the universal (finite) etale cover of $\bar{X}$. Note that $\tilde{\bar{X}}\to \bar{X}\to X$ becomes the universal (finite) etale cover of $X$. Then there exists a (finite) etale cover $Y\to X$ such that $\bar{Y}\to \bar{X}$ is exactly same with $\tilde{\bar{X}}\to \bar{X}$.

First of all, I am not sure whether all the conditions in the above statement are necessary (for example, geometrically irreducibility, smoothness, finiteness of covers). I would like to know the most general form of the theorem.

Second, I don't know the definition of universal (finite) etale covering. In fact, I know one definition of it which describes it by an inverse limit of some good (finite) etale covers. But I am not sure whether it is most common.

If someone can give me an exact statement, I would most appreciate it. However, anything similar would also be very helpful for me.