Suppose that $X, B$ are smooth irreducible varieties over $\mathbb{C}$ and $f : X \to B$ is a smooth proper morphism. Then we can consider the homotopy exact sequence:
$$ \pi_2(B) \to \pi_1(F) \to \pi_1(X) \to \pi_1(B) \to 1 $$
I am looking for a situation where the map $\pi_2(B) \to \pi_1(F)$ is NOT zero.
Some remarks:
(1) if $B$ is a curve this cannot happen since either $B = \mathbb{P}^1$ or $\pi_2(B) = 0$. In the first case, it is a theorem that every smooth proper map $X \to \mathbb{P}^1$ has a section so the sequences are split.
(2) I don't think that $X \to B$ can be a fibration in either curves or abelian varieties since the connecting map factors through $\pi_2(\mathcal{M}_g(\mathbb{C}))$ (resp. $\pi_2(\mathcal{A}_g(\mathbb{C}))$) both of which are zero for $g > 0$.
(3) Of course there are examples if we drop properness. Eg the ``algebraic hopf fibration''
$$ \mathbb{A}^2 \setminus \{ 0 \} \to \mathbb{P}^1 $$
My second question deals with the cases where the connecting map is zero. If $\pi_2(B) \to \pi_1(F)$ is zero, is it true that etale homotopy sequence is also short exact,
$$ 1 \to \pi_1^{\text{et}}(F) \to \pi_1^{\text{et}}(X) \to \pi_1^{\text{et}}(B) \to 1 $$
Some remarks:
(1) this is true if $f : X \to B$ has a section
(2) therefore it is true when $B$ is a curve since either $B = \mathbb{P}^1$ in which case there is a section or $B$ is both a topological and etale $K(\pi, 1)$.
(3) it is not clear to me what happens for fibrations in curves or abelian varieties. Indeed, I have heard that $\mathcal{A}_g$ is NOT an etale $K(\pi, 1)$ even though it is a topological $K(\pi, 1)$ so maybe a family of abelian varieties could give a counterexample.
If you knew the following facts then the proof of the etale homotopy SES would be immediate:
(1) there is a long exact sequence of higher etale homotopy groups
(2) the connecting maps are functorial giving a morphism of long exact sequences,
$$\require{AMScd} \begin{CD} \pi_2(B) @>>> \pi_1(F) @>>> \pi_1(X) @>>> \pi_1(B) @>>> 1 \\ @VVV @VVV @VVV @VVV \\ \pi_2^{\text{et}}(B) @>>> \pi_1^{\text{et}}(F) @>>> \pi_1^{\text{et}}(X) @>>> \pi_1^{\text{et}}(B) @>>> 1 \end{CD}$$
(3) that the maps $\pi_i(F) \to \pi_i^{\text{et}}(F)$ induce isomorphisms $\widehat{\pi_i(F)} \cong \pi_i^{\text{et}}(F)$.
My guess is that (1) and (2) are true but that (3) fails for $i > 1$. Does anyone point me to a reference or a counterexample in the case $i = 2$.
