On abelian spaces

Question

By Allen Hatcher's book, a space is called abelian if it has trivial action of $\pi_1$ on all homotopy groups $\pi_n$, since when $n=1$ this is the condition that $\pi_1$ be abelian.

My quesion is that:

1. Is $S^1 \vee S^2$ an abelian polyhedron?

2. Is there an abelian polyhedron $P$ with abelian $\pi_1 (P)$ and infinitely generated $H_i (\tilde{P},\mathbb{Z})$ for some $i\geq 2$?