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 $G$, is 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 arise 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 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.