  Let $G$ be a finitely presented group that can be written as an extension \[
1 \longrightarrow K \longrightarrow G \longrightarrow F_2 \longrightarrow 1 \]
where $K$ is either the finitely generated free group $F_n$, $n > 2$ or the
fundamental group of a closed surface of genus $g > 1$. We prove that if the
image of the monodromy map $\rho \colon F_2 \to \operatorname{Out(K)}$ contains
an element $\varphi \in \operatorname{Out(K)}$ such that the mapping torus $K
\rtimes_{\varphi} \Bbb{Z}$ is virtually residually finite rationally solvable
(for instance whenever the mapping torus is hyperbolic), then $G$ is not
coherent. This applies, in particular, when the image is a purely
pseudo--Anosov free subgroups of the mapping class group.
