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 extensions of (free) groups and the motivations behind such studies. More specifically, we will give necessary and sufficient conditions for a free extension of a (non-Abelian) free group given by a subgroup of Out(F_n) to be hyperbolic and relatively hyperbolic, in the most general case for the subgroup. We will also talk about hierarchical hyperbolicity of certain free-by cyclic groups. Based on joint works with Ghosh, and Bongiovanni, Ghosh and Hagen.