My question is: what is the extension class of a nilpotent group?
I'm trying to understand the proof sketch of Theorem 6.2 of Morita's paper "A Linear Representation of the Mapping Class Group." I don't have a link to the paper, but hopefully I give enough context below.
I have a representation $$f: G \rightarrow (A \times B) \rtimes C$$
This representation restricts to $$f|_{H} : H \rightarrow A \times B$$ and projects to $$g: H \rightarrow A$$
Ultimately, the goal is to find $\text{ker}(g^*: H^2(H) \rightarrow H^2(A))$. As a first step, Morita claims the extension class of the nilpotent group $A \times B$ can be expressed as a homomorphism $\wedge^2 B \rightarrow \wedge^2 A$ and that this is nontrivial. The rest of the proof relies on this fact.
Can you help me understand what this means and maybe share some simple examples of what this tells us?