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Using the train track machinery of Bestvina-Feighn-Handel and Handel- Mosher,we give a necessary and sufficient condition for a free extension of a freesubgroup of $Out(\mathbb{F})$ generated by exponentially growing outerautomorphisms to be cusp preserving (strongly) relatively hyperbolic. Usingweak attraction theory, we investigate fixed points of exponentially growingautomorphisms on the complex of free factors and we prove a necessary conditionfor non-relative hyperbolicity of such free by free extensions. Moreover, thisinteraction leads us to a new necessary and sufficient condition for such anextension to be hyperbolic; and as a result, we suggest an alternativedefinition(and a subsequent generalization) of convex cocompactness in$Out(\mathbb{F})$, given only by the lamination sets of exponentially growingouter automorphisms.

Hyperbolicity and relative hyperbolicity of free extensions of free groups