$X$ a manifold. A short exact sequence of finite abelian groups $$ 1\rightarrow A_1\rightarrow A_2\rightarrow A_3\rightarrow 1 $$ induces a long exact sequence of cohomology groups of $X$ with coefficients, whose connecting map is the Bockstein $\beta : H^n(X,A_3)\rightarrow H^{n+1}(X,A_1)$. The simplest definition I know is algebraic: you lift $\alpha \in H^n(X,A_3)$ to an $A_2$ valued cochain, which is not closed but the differential takes value in $A_1$ and this is are representative of $\beta(\alpha)$.
I read (for instance in https://ncatlab.org/nlab/show/Bockstein+homomorphism) that there is an alternative geometric definition. From the SES one associates a long fibre sequence of classifying spaces $$ BA_1\rightarrow BA_2\rightarrow BA_3\rightarrow B^2A_1\rightarrow ... $$
First question: how one construct this sequence? Then the Bockstein can be constructed as follows. One represents $\alpha \in H^n(X,A_3)$ with a map $X\rightarrow B^nA_3$, and composing it with $B^nA_3\rightarrow B^{n+1}A_1$ one gets the element in $H^{n+1}(X,A_1)$.
Appartently this is also the Bocksetin: how can we see this?