  We introduce the co-surface graph $\mathcal{CS}$ of a finitely generated free
group $\mathbb{F}$ and use it to study the geometry of hyperbolic group
extensions of $\mathbb{F}$. Among other things, we show that the Gromov
boundary of the co-surface graph is equivariantly homeomorphic to the space of
free arational $\mathbb{F}$-trees and use this to prove that a finitely
generated subgroup of $\mathrm{Out}(\mathbb{F})$ quasi-isometrically embeds
into the co-surface graph if and only if it is purely atoroidal and
quasi-isometrically embeds into the free factor complex. This answers a
question of I. Kapovich. Our earlier work [Hyperbolic extensions of free
groups, arXiv:1406.2567] shows that every such group gives rise to a hyperbolic
extension of $\mathbb{F}$, and here we prove a converse to this result that
characterizes the hyperbolic extensions of $\mathbb{F}$ arising in this manner.
As an application of our techniques, we additionally obtain a Scott--Swarup
type theorem for this class of extensions.
