Given a finitely generated subgroup Γ≤Out(𝔽) of the outer automorphism group of the rank r free group 𝔽=Fr, there is a corresponding free group extension 1→𝔽→EΓ→Γ→1. We give sufficient conditions for when the extension EΓ is hyperbolic. In particular, we show that if all infinite order elements of Γ are atoroidal and the action of Γ on the free factor complex of 𝔽 has a quasi-isometric orbit map, then EΓ is hyperbolic. As an application, we produce examples of hyperbolic 𝔽-extensions EΓ for which Γ has torsion and is not virtually cyclic. The proof of our main theorem involves a detailed study of quasigeodesics in Outer space that make progress in the free factor complex.