$Q_nS^0$ is the $n$-th component of infinite loop space, in other words $Q_nS^0=\varinjlim_k(\Omega^kS^k)_n$, where $(\Omega^kS^k)_n$ refers to homotopy classes of maps $(S^k,\ast)\to (S^k,\ast)$ of degree $n$.
$H_{\ast}(Q_nS^0,\mathbb{Z}/2)$ is a comodule over mod 2 dual Steenrod algebra $\mathcal{A}_{\ast}$, since for two spectra $E$ and $X$, the map $$E\wedge X=E\wedge S^0\wedge X\to E\wedge E\wedge X$$ induces $$E_{\ast}(X)\to\pi_{\ast}(E\wedge E\wedge X)=E_{\ast}(E)\otimes_{\pi_{\ast}(E)}E_{\ast}(X)\ \ (\ast)$$ Let $E$ be the Eilenberg–Maclane spectrum $H\mathbb{Z}/2$ and $X$ be the spectrum generated by $Q_nS^0$.
Question: How to obtain precise formula of $H_{\ast}(Q_nS^0)\rightarrow H_{\ast}(Q_nS^0)\otimes \mathscr{A}_{\ast}$ from $(\ast)$ ?
Is it a simple thing or are there some references about it ?
