Using pairing in Atiyah-Hirzebruch spectral sequence one can show that homology of $BU(n)$ is a free abelian group with basis $\alpha_{k_1}\cdots\alpha_{k_t}$, $k\leqslant n$, where $\alpha_{i} = \big((c_1)^i\big)^* \in H_{2i}(\mathbb C P^\infty)$. Therefore, passing to (co)limits and applying the Thom isomorphism, we have \begin{equation} H_*(MU) \cong \mathbb Z[a_1, a_2, \ldots]. \end{equation}
Since polynomial generators come as image of homology classes of $\mathbb C P^\infty$, $\mathfrak A_p^*$-comodule structure of $H_*(MU; \mathbb F_p)$ can be deduced from the one of $H_*(\mathbb C P^\infty; \mathbb F_p)$.
Is there any similar description of $H_*(MSU; \mathbb F_p)$?
UPD.
Let $S = \mathbb F_p[x_n| n+1\neq p^t]$. Define $f\colon H_*(MU;\mathbb F_p) \to S$ by the formula \begin{equation*} f(a_n) = \begin{cases} x_n;\quad&\text{if $n+1 \neq p^t$;} \\ 0;&\text{otherwise.} \end{cases} \end{equation*} Then the composition \begin{equation*} H_*(MU;\mathbb F_p) \xrightarrow{\text{coaction}} \mathfrak A_p'\otimes_{\mathbb F_p} H_*(MU;\mathbb F_p) \xrightarrow{\mathrm{id}\otimes f} \mathfrak A_p'\otimes_{\mathbb F_p} S \end{equation*} is an isomorphism of $\mathbb F_p$-algebras and $\mathfrak A_p^*$-comodules. Here $\mathfrak A_p' = \mathbb F_p[\xi_1, \xi_2, \ldots]$ is the quotient of the dual Steenrod algebra $\mathfrak A_p^*$.
The Adams' paper Primitive Elements in the $K$-theory of $BSU$ contains a computation of homology $H_*(BSU; \mathbb Z)$. Namely, $H_*(BSU; \mathbb Z) \cong \mathbb Z[y_2, y_3, \ldots]$, $\deg y_i = 2i$. And therefore, $H_*(MSU; \mathbb Z) \cong \mathbb Z[Y_2, Y_3, \ldots]$, $\deg Y_i = 2i$. But still the proof does not describe the embedding \begin{equation*} H_*(MSU; \mathbb Z) \hookrightarrow H_*(MU; \mathbb Z). \end{equation*}
Are there any explicit formulae for the images of $y_n$ in $H_*(MU)$? I would be really grateful for something explicit that could be useful for describing the $\mathfrak A_p^*$-comodule structure of $H_*(MSU;\mathbb F_p)$.