In the book The homology of iterated loop spaces, the homology Hopf algebra

(1) $$ H_*(\Omega^n \Sigma^n X;\mathbb{Z}_p) $$

for primes $p\geq 2$ is obtained on p. 226, Thm. 3.2. In particular, the homology Hopf algebra

(2) $$ H_*(\Omega^nS^n;\mathbb{Z}_2) $$ is known. However, when I use the Cartan formula and Adem relations to compute the coproduct of (2), I find it is quite complicated when modulo the Adem relations, and do not know how to get the explicit expression of the coproduct.

Question: Are there any references where I can find the explicit expression of the cohomology algebra $$ H^*(\Omega^nS^n;\mathbb{Z}_2)? $$

I obtain that $\Omega S^1=\text{Map}_*(S^1,S^1)\simeq [S^1;S^1]_*=\pi_1(S^1)=\mathbb{Z}$. Hence $H^*(\Omega S^1;\mathbb{Z}_2)=\oplus_{k\in\mathbb{Z}} \mathbb{Z}_2a_k$, $|a_k|=0$. How to compute the following examples

$$ H^*(\Omega^2S^2;\mathbb{Z}_2) $$ and $$ H^*(\Omega^3S^3;\mathbb{Z}_2)? $$