Let $k'/k$ be an extension of algebraically closed fields of characteristic $0$, and $X$ a concentrated (i.e. quasi-compact and quasi-separated) scheme over $k$.

**Question**: is the pullback functor from finite étale covers of $X$ to finite étale covers of $X_{k'}$ an equivalence?

If $X$ is connected, then by SGA1, Exposé XIII, Proposition 4.6b we know that $\pi_1(X)=\pi_1(X_{k'})$, and hence the answer is yes. If the connected components of $X$ are open (for example if $X$ is of finite type), then we can reduce ourselves to the connected case, but in general connected components are not open.