It's well known that $\pi_1^{èt}($Spec$(\mathbb{F}_p))=\hat{\mathbb{Z}}$, for $p$ a prime. However, the scheme $$X_{p,q}:=\text{Spec}(\mathbb{F}_p)\times_{\text{Spec}(\mathbb{Z})}\text{Spec}(\mathbb{F}_q)$$ is empty if $p\neq q$, and isomorphic to Spec$(\mathbb{F}_p)$ if $p=q$, so neither work. The idea was to use the natural map, $$\pi_1^{èt}(X\times_{S}Y)\rightarrow \pi_1^{èt}(X)\times \pi_1^{èt}(Y)$$ somehow. So does such a scheme exist?