I have a follow-up question to my question here: How are simple groups the building blocks?
In that question I asked about what it means when we say that the simple (finite) groups are the building blocks of all finite groups. It seems it all comes down to understanding that if you have a short exact sequence: $$ 1 \to H \to G \to G / H \to 1 $$ then $G$ us built out of $H$ and $G /H$. From a comment to an answer the point is made that that there are various ways that one can say that $G$ is built out of $H$ and $G/H$.
My follow-up question is what this means? What are examples of what can be said? How can, for example, can information about $G$ be "lifted" from $H$ and $G/H$?