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Is $\pi_2 (X_i)$ a free $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,2$?
Let $X_1$ be the suspension of $\mathbb{R}P^2$ and $X_2=\bigvee_{1\leq i\leq n} (\vee_{r_i} \mathbb{S}^i)$.
Is $\pi_2 (X_i)$ a projective (or a free) $\mathbb{Z}\pi_1 (X_i)$-module for $i=1,...
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