This is an assignment question with $3$ steps as follows:
$1.$ Prove that $H^2(K(\mathbb{Z},2);\mathbb{Z}/2)\cong \mathbb{Z}/2.$ Use this generator to obtain a map $K(\mathbb{Z},2)\rightarrow K(\mathbb{Z}/2,2)$. Prove that the homotopy fibre of the map is $\simeq$ $K(\mathbb{Z}/2)$.
$2.$ In the above fibration $\big(K(\mathbb{Z},2)\xrightarrow{f}{} K(\mathbb{Z},2)\rightarrow K(\mathbb{Z}/2,2)\big),$ prove that we get an induced map $q:Cone (f)\rightarrow K(\mathbb{Z}/2,2).$ Prove that $q_*$ is surjective on homotopy groups.
$3.$ Prove that $q$ is an $4$-equivalence. Use this to deduce that $H^4(K(\mathbb{Z}/2,2))\cong\mathbb{Z}/2$.
I have solved the first two parts of the question but I am completely clueless about the last part. I don't know how to calculate the homotopy groups of $Cone(f)$, I guess there must be a way around it. Any help is appreciated.