I am interested in the following question which I asked at MSE and did not get any clues. How to find three two-dimensional CW complexes $K_1, K_2, K_3$ with non-trivial $\pi_2$ and injective embeddings $f:K_1\rightarrow K_2, g:K_2\rightarrow K_3$, such that the induced homomorphisms $f_*, g_*$ on the level of $\pi_2$ would be trivial?

The question with one embedding is relatively simple. Consider $K_1 = \mathbb{R}P^2, K_2 = \mathbb{R}P^2\cup C$, where $C$ is the 2-dimensional cell which kills $\pi_1$. Then embedding $f: \mathbb{R}P^2 \rightarrow \mathbb{R}P^2\cup C$ has the property that $f_* = 0$. This follows from Hurewicz theorem and the fact that $H_2(\mathbb{R}P^2)=0$.

The problem and the example are taken from R. Mikhailov lectures. This problem is tied with Whitehead asphericity conjecture.

Also, any references which could help to feel this kind of algebraic topology deeper are welcome.