Group extensions (question about definitions)

In summary, the discussion on group extensions revolves around the definitions and properties of extending groups. It addresses the conditions under which a group can be considered an extension of another group, exploring concepts such as normal subgroups, quotient groups, and the relationships between groups in the context of exact sequences. The focus is on clarifying the terminology and providing a framework for understanding how groups can be built from simpler components.
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Korybut
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TL;DR Summary
Definition question
Hello!

I would like to be sure about my understanding of the definition provided in screenshot below

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1. What is this ##\mathcal{E}(G,N)##? I know that not all extension are isomorphic so I wonder What are the elements of ##\mathcal{E}## groups? Or maybe all Es diffeomorphic to each other. Don't know...

2. Please explain how ##u^*## is the pullback of ##u:G^\prime \rightarrow G##

3. Very new to the subject so I would be grateful if someone recommends nice manual explainig basics.

Many thanks in advance
 

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