Relative hyperbolicity of free extensions of free groups

We give necessary and sufficient conditions for a free-by-free group to be relatively hyperbolic with a cusp-preserving structure. Namely, if ${\phi }_1,\dots ,{\phi }_k$ is a collection of exponentially growing outer automorphisms with a common invariant \emph{subgroup system} such that any conjugacy class in the complement of this system grows exponentially under iteration by all ${\phi }_i$, then such a subgroup system can be used to construct a collection of peripheral subgroups relative to which, the extension of $\mathbb{F}$ by the free group generated by sufficiently high powers of ${\phi }_1,\dots ,{\phi }_k$, will be hyperbolic.