I am reading Spanier's algebraic topology book and he states that the quotient of $\pi_n(X,A,x_0)$ by the action of $\pi_1(X,A,x_0)$ is Abelian. I think there is a typo here somewhere as for proof he references a lemma 7.3.12 which deals explicitly with $\pi_2$, not $\pi_n$ for arbitrary $n$, using a proof that really does not seem to generalize well at all (although perhaps there's some inductive bootstrapping using an isomorphism $\pi_n(X,A)\cong \pi_{n-1}(\Sigma X, \Sigma A)$ or something of this ilk that I'm missing.)
Is the statement true? Is there a proof of it?