If we have a variety, $X$, over a field, $k$, and $x$ is a geometric point of $X$, and let $\bar x$ be a geometric point of $X_{k^s} := X \times_k k^s$ above $x$ then we have the following short exact sequence:

$1 \rightarrow \pi_1(X_{k^s}, \bar x) \rightarrow \pi_1(X,x) \rightarrow Gal(k) \rightarrow 1$

Implicit in this is a choice of $k^s$ (if you want, this is a choice of geometric point, $z$, on $Spec(k)$; $\pi_1(Spec(k), z)=Gal(k)$).

I'm wondering how to interpret the splitting of this short exact sequence, and more specifically: what is the significance of choosing different splittings? I'm having a hard time picturing intuitively how to think of this splitting.