The interest in the geometry of group extensions started with the geometrization theorem of Thurston for compact irreducible atoroidal 3-manifolds. We will talk about the geometry of group extensions and the motivations behind such studies in the cases of closed surface groups and free groups. More specifically, we will talk about the most general case so far and, we will give necessary and sufficient conditions for a free extension of a (non-Abelian) free group given by a subgroup of the outer automorphism group of the free group (Out(F_n)) to be hyperbolic and relatively hyperbolic. Joint work with Pritam Ghosh.