Question:Normal subgroup quotient group determine whole group counterfactual

This is a follow-up question to Question:Normal subgroup quotient group determine whole group.

Q: What would happen to group theory if it were true that the isomorphism type of a group is completely determined by knowing the isomorphism type of the normal subgroup and quotient group?

A: There wouldn't be much of group theory in that case. All groups would be semisimple in the sense of being direct products of the simplest constructs: the groups of prime order. In other words, they would just be direct products of elementary abelian groups for various primes. In particular, every group would be abelian. There would be no scope for building complicated structures.