Thanks to a suggestion by @Igor Belegradek, I am interested also in a simpler problem of this earlier question 301523, by knowing what can we say about the classification of fibrations for classifying spaces $B^2M \equiv B^2\mathbb{Z}_2$ and $BG \equiv BSO(2)$ or $BO(2)$.
Here we can take either:
- $B^2M=B^2\mathbb{Z}_2$ as fibers, and $BG=BSO(2)$ (or $BO(2)$) as base space.
or
- $BG=BSO(2)$ (or $BO(2)$) as fibers, and $B^2M=B^2\mathbb{Z}_2$ as base space.
For example,
(A) We consider the Postinikov classes $\omega \in H^3(BG,M)$. This classifies the fibrations of $$ B^2M \hookrightarrow BG_\text{{new}} \to BG. $$ If I understand correctly, the distinct $BG_\text{{new}}$ are determined by the classes of cocycles $\omega \in H^3(BG,M)$.
(B) On the other hand, we have $1\to \mathbb{Z}_2 \to Spin(2) \to SO(2) \to 1$, this gives rise $B\mathbb{Z}_2 \to BSpin(2) \to BSO(2) \to B^2\mathbb{Z}_2,$ $$ BSpin(2) \to BSO(2) \to B^2\mathbb{Z}_2, $$ which seems to suggest a fibration of $BSpin(2)$ over $B^2\mathbb{Z}_2$
(C) On the other hand, we further have $1\to SO(2) \to O(2) \to \mathbb{Z}_2 \to 1$, this gives rise $BSO(2) \to BO(2) \to B \mathbb{Z}_2,$ which seems not to suggest any related fibration pertinent to my question?
My questions:
(a) Do (B) and (C) above really suggest a valid fibration of classifying space fibered over another classifying spaces? How do we classify them? Are there similar classifications like Postinikov classes?
(b) The first example (A) and the later (B) and (C) look very different from each other. Are there any relations or generalizations to relate each of them?
(c) In general, given $B^2M \equiv B^2\mathbb{Z}_2$ and $BG \equiv BSO(2)$ or $BO(2)$, what are other possible fibrations between them (these classifying spaces)?
