For a pointed space $X$ and an Abelian group $G$, the loop-space functor induces a homomorphism $\omega:H^n(X,G)\to H^{n-1}(\Omega X,G)$. More concretely, $\omega$ is given by the Puppe sequence

$$\cdots \to \Omega X \overset{\omega f}{\to} K(G,n\!-\!1) \to Mf \to X \overset{f}{\to} K(G,n)\,.$$

Equivalently, $\omega$ can also be defined as the pullback of the evaluation map $\Sigma\Omega X {\to} X$, i.e.,

$$[X, K(G,n)]\overset{\omega}{\to}[\Sigma\Omega X, K(G,n)]\simeq[\Omega X, K(G,n\!-\!1)]\,.$$

Q1. Where can I find a discussion on $\omega$'s properties, e.g., kernel and image?

Q2. What's the relationship between $\omega$ and differentials in the Serre spectral sequence associated to $\Omega X\to *\to X$?