After defining convex cocompact subgroups of the mapping class group, Farb and Mosher asked if every finitely generated and purely pseudo-Anosov subgroup had to be convex cocompact. Combined with work of Hamenstädt, either answer to this question has strong implications for the geometry of surface group extension. We will introduce the connections between convex cocompactness and the hyperbolicity of surface extensions, then discuss joint work with Chris Leininger where we prove that the answer to Farb and Mosher's question is "yes" for subgroups of non-hyperbolic fiber 3-manifold groups that include into the mapping class group via the Birman Exact sequence. Combined with earlier work of Kent--Leininger--Schleimer and Dowdall--Kent-Leininger, this completes Farb and Mosher's question for these 3-manifold groups.