The next two claims completely describe $H_*(\Omega^2S^3;\mathbb{Z})$. This follows from several sources. For example, from already mentioned in the answer of Nicholas Kuhn book of Joe Neisendorfer.
Theorem 1. The space $H_*(\Omega^2S^3;\mathbb{Z}_p)$ is a primitively generated Hopf algebra such that $$ H_*(\Omega^2S^3;\mathbb{Z}_p)= \begin{cases} \Lambda_p[x_0,x_1,x_2,\cdots]\bigotimes\mathbb{Z}_p[y_0,y_1,y_2,\cdots] &\quad\mbox{for}\quad p>2,\\[1mm] \mathbb{Z}_2[x_0,x_1,x_2,\dots]&\quad\mbox{for}\quad p=2, \end{cases} $$ where $\deg(x_r)=2p^r-1,\,\deg(y_r)=2p^{r+1}-2$. In particular \begin{eqnarray*} \sum_{q=0}^\infty\dim H_q\big(\Omega^2S^3;\mathbb{Z}_p\big)\,t^q&=& \prod_{r=0}^\infty\frac{1+t^{2p^r-1}}{1-t^{2p^{r+1}-2}}\qquad\text{for $p>2$},\\ \sum_{q=0}^\infty\dim H_q\big(\Omega^2S^3;\mathbb{Z}_2\big)\,t^q&=& \prod_{r=0}^\infty\frac{1}{1-t^{2^{r+1}-1}}\,. \end{eqnarray*}
Theorem 2. There is an isomorphism of $\mathbb{Z}$--modules $$ H_q(\Omega^2S^3;\mathbb{Z})= \begin{cases} \mathbb{Z}&\text{for $q=0,1$},\\[1mm] \bigoplus_{p\geqslant 2}\beta_p(H_{q+1}(\Omega^2S^3;\mathbb{Z}_p))&\text{for $q\geqslant 2$}, \end{cases} $$ where $\beta_p:H_{q+1}(\Omega^2S^3;\mathbb{Z}_p)\longrightarrow H_q(\Omega^2S^3;\mathbb{Z})$ is the Bockstein homomorphism corresponding to the exact sequence of coefficients $0\longrightarrow\mathbb{Z}\stackrel{\times p}\longrightarrow\mathbb{Z}\longrightarrow\mathbb{Z}_p\longrightarrow 0$ for prime $p\geqslant 2$. Homomorphisms $\beta_p$ are the graded injective differentiations. The action of $\beta_p$ is defined by the formulas $$ b_2(x_r)=x^2_{r-1},\qquad b_p(x_0)=0,\qquad \begin{cases} b_p(x_r)=y_{r-1},\\ b_p(y_r)=0 \end{cases} \quad\text{for $p>2$}, $$ where $r\geqslant 1$.