On the group structure of [Ω mathbb S², Ω Y]

Let $J(X)$ denote the James construction on a space $X$ and $J_n(X)$ be the $n$-th stage of the James filtration of $J(X)$. It is known that $[J(X),\Omega Y]\cong \lim\limits_{\leftarrow} [J_n(X),\Omega Y]$ for any space $Y$. When $X=\mathbb S^1$, the circle, $J(\mathbb S^1)=\Omega \Sigma \mathbb S^1=\Omega \mathbb S^2$. Furthermore, there is a bijection between $[J(\mathbb S^1),\Omega Y]$ and the product $\prod_{i=2}^\infty \pi_i(Y)$, as sets. In this paper, we describe the group structure of $[J_n(\mathbb S^1),\Omega Y]$ by determining the co-multiplication structure on the suspension $\Sigma J_n(\mathbb S^1)$.