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 $r\geqslant 1$.