$\DeclareMathOperator\SU{SU}$For the $\mod p$ singular cohomology of classical Lie groups, such as $H^*(\SU(n); \mathbb{Z}/p\mathbb{Z})$, there are well known results about the actions of the stable cohomology operations, say the Steenrod operator $P^k$ on it.
I was wondering if there are any similar results about general cohomology theories. To put it more precise, for a classical Lie group $G$, are there any explicit formulas for the following homomorphisms
\begin{equation*}
\phi: E^*E\otimes_{E_*} E^*(G)\rightarrow E^*(G)
\end{equation*}
or dually
\begin{equation*}
\psi: E_{*}(G)\rightarrow E_{*}E \otimes_{E_*} E_{*}(G).
\end{equation*}
For example, if we take $E$ to be the Morava $K$-theory $K(n)$, and $G$ to be $\SU(m)$, then the question is whether there are explicit formulas for the following homomorphisms
\begin{equation*}
\phi: K(n)^*{K(n)}\otimes_{K(n)_*} K(n)^*(\SU(m)\rightarrow K(n)^*(\SU(m))
\end{equation*}
or dually
\begin{equation*}
\psi: K(n)_{*}(\SU(m))\rightarrow K(n)_{*}K(n) \otimes_{K(n)_*} K(n)_{*}(\SU(m)).
\end{equation*}
This question arises when considering the homotopy set $[G_1,G_2]$ for Lie groups $G_1$ and $G_2$. While this homotopy set is hard to compute, it is worthwhile to consider the Boardman map
\begin{equation*}
B: [G_1,G_2]\rightarrow \mathrm{Hom}_{E^*E}(E^*(G_2),E^*(G_1))
\end{equation*}
for some cohomology theory $E$. And the homomorphism $\phi$ as mentioned before is indispensable for the calculation of the Hom-set.
