  Let $S$ be a connected Dedekind scheme and $X$ an $S$-scheme provided with a
section $x$. We prove that the morphism of fundamental group schemes
$\pi_1(X,x)^{ab}\to \pi_1(\mathbf{Alb}_{X/S},0_{\mathbf{Alb}_{X/S}})$ induced
by the canonical morphism from $X$ to its Albanese scheme $\mathbf{Alb}_{X/S}$
(when the latter exists) fits in an exact sequence of group schemes $0\to
(\mathbf{NS}^{\tau}_{X/S})^{\vee}\to \pi_1(X,x)^{ab}\to
\pi_1(\mathbf{Alb}_{X/S},0_{\mathbf{Alb}_{X/S}}) \to 0$ where the kernel is a
finite and flat $S$-group scheme. Furthermore we prove that any finite and
commutative quotient pointed torsor over the generic fiber $X_{\eta}$ of $X$
can be extended to a finite and commutative pointed torsor over $X$.
