# exact sequence of fundamental groups associated to “almost” smooth families of curves

Suppose I have a proper, flat family of curves $X \to S$ that has a section. Fix a basepoint $s \in S$ and let $X_s$ denote the corresponding fiber. Let $\mathbb{L}$ be a set of primes which does not contain any of the residue characteristics of $S$. Let $\pi_1(X_s)^{\mathbb{L}}$ be the pro-$\mathbb{L}$ quotient of the etale fundamental group of $X_s$ (here for convenience I'm ignoring basepoints), and define $\pi_1(X)'$ as follows: if $N$ is the largest normal subgroup of the kernel $K$ of $\pi_1(X) \to \pi_1(S, s)$ such that $K/N$ is pro-$\mathbb{L}$, then $\pi_1(X)' = \pi_1(X) / N$. Then clearly we have maps $\pi_1(X_s)^{\mathbb{L}} \to \pi_1(X)' \to \pi_1(S, s)$ whose composition is trivial. Now a result of Grothendieck (SGA1, XIII.4.3) states that, under certain technical conditions on the morphism $X \to S$ (including being 0-acyclic and locally 1-aspheric), these maps form a short exact sequence $$1 \to \pi_1(X_s)^{\mathbb{L}} \to \pi_1(X)' \to \pi_1(S, s) \to 1.$$ (See When can the "homotopy exact sequence" of etale fundamental groups for a smooth curve fail to be exact? for a similar question regarding this result.) It is known that smoothness of $X \to S$ is enough to imply these technical conditions (SGA1, XIII.4.4).

What I want to ask, before trying to work my way through all the technicalities in understanding what kinds of properties imply and are implied by 0-acyclic and 1-aspheric, is this: Does anyone know for what kinds of non-smooth families $X \to S$ does this short exactness still hold? In particular, do we still get short exactness if $X \to S$ has a semistable fiber $X_t$ with no vanishing cycles, i.e. such that the Jacobian of $X_{\eta}$ (where $t \in S$ is a specializataion of $\eta \in S$) has good reduction at $t$? (From what I do understand of the definitions of 0-acyclic and 1-aspheric, these are conditions involving cohomology of sheaves of abelian $\mathbb{L}$-groups, which seems vaguely promising since such mild degeneracy shouldn't be visible in cohomology, yes?)