Recall that the space $A$ is homotopy dominated by $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ such that $gf\simeq id_A$.

Suppose that $X$ is a wedge of some spheres and $A$ homotopy dominated by $X$. Let $K$ be a bouquet of circles so that map $\phi :K\longrightarrow A$ induces an isomorphism on fundamental groups. Put $\pi_2 (\phi )=\pi_2 (M,K\times 1)$ where $M$ is the mapping cylinder of $\phi$. I know that $\pi_2 (\phi )$ is a finitely generated $\mathbb{Z}\pi_1 (A)$-module. My question is that: how do I prove $\pi_2 (\phi )$ is free $\mathbb{Z}\pi_1 (A)$-module?

Thanks in advance.