TITLE: Is the mapping torus of an automorphism of a free group virtually an amalgamated product?
QUESTION [5 upvotes]: Let $F$ be a nonabelian finitely generated free group, 
let $\tau \in \mathrm{Aut}(F)$ be an element of infinite order, 
and set $G = F \rtimes \mathbb{Z}$, 
where the action of $\mathbb{Z}$ on $F$ is given by $\tau$.
Must there be a finite index subgroup $U$ of $G$, 
such that $U \cong A*_CB$ where $C$ is a proper subgroup of both $A$ and $B$, and its index in either $A$ or $B$ is at least $3$?
What happens if $\tau$ is inner?

REPLY [3 votes]: If you drop "virtually" from your question, then this has been analyzed very thoroughly by Brinkmann. Whether this is directly relevant to your question is, of course, open to debate.
Peter Brinkmann, MR 1934698 Splittings of mapping tori of free group automorphisms, Geom. Dedicata 93 (2002), 191--203.