Computing the homotopy groups $\pi_n(G,G_q)$

Question

Let $G_q$ be the space of degree one maps $S^{q-1}\rightarrow S^{q-1}$ and $G=lim G_q$ under suspensions $G_q\subset G_{q+1}$. What are the known computations of $\pi_n(G,G_q)$?
I found that $\pi_3(G,G_3)=\mathbb{Z}_2$ and $\pi_4(G,G_3)=\mathbb{Z}$ in Haefliger's https://doi.org/10.2307/1970475 section 5.16.

Answer 1

By Theorem A of Milgram's On the Haefliger knot groups, there is a $(2k-3)$-connected map

$$\mathrm{SO}/\mathrm{SO}(k) \longrightarrow \mathrm{G}/\mathrm{G}_k,$$

and the left side is well-studied in the metastable range (see the work of Mahowald).