I was trying to undetstand the statement that vanishing of second Stiefel whitney class implies the existence of Spin structure. I came across the answer https://math.stackexchange.com/a/808396/474870. I have a question related to this.
Consider the exact sequence $Z_2\to Spin(n)\to SO(n)$. Since $Spin(n)$ is a $Z_2$ bundle, we have a map $\phi: SO(n) \to BZ_2$. According to the cited answer above, this induces a map $B\phi: BSO(n)\to B^2Z_2$, where $B^2Z_2$ is defined as $\Omega(B^2Z)=BZ_2$. I was wondering how to define this map $B\phi$.
I know how to define the map $B\rho : BG\to BK$ corresponding to a group Homomorphism $G\to K$. As discussed in https://sites.math.washington.edu/~mitchell/Notes/prin.pdf, $B\rho$ is the classifying map for the $K$-bundle $(EG \times K)/G \to BG$. But I have no idea, how to replicate this construction to the case in question $\phi: SO(n) \to BZ_2$. It will be really helpful if someone can guide me with this issue. I will also appreciate if someone can suggest a good reference to learn this sort of stuff.
