The connection to group extensions is much more subtle, and not entirely easy to tie in with all these much more fundamental topological questions.

As I mentioned in another answer, much of homological algebra is about using the techniques from algebraic topology, but in situations where the geometrical underpinnings vanish from sight.

Basically, it turns out that the process of going from, say, a triangulated space to its homology groups is by building chain complexes. And finding chain complexes that replace the space, retaining some of its properties.

So, for studying group extensions, it turns out that doing something similar - constructed either by building a topological space out of the group, or by building chain complexes directly through a somewhat ad hoc construction - we can say things about the group. But studyin group extensions through group cohomology is much easier to understand by approaching it with studying how certain functors deal with short exact sequences, than by trying to frame it in terms of geometry and geometrical topology.

The connection, really, appears in that in both cases, the way to find an answer is through replacing the original object with a chain complex, and then doing Stuff, including finding homology groups, to that chain complex.