I am currently working through the paper ''The Homotopy of $L_2V(1)$ for the Prime $3$" by Goerss, Henn, Mahowald which can be found in the book Categorical Decomposition Technique in Algebraic Topology.
On page 136 they make reference to the map $$\lambda_X \colon \text{Ext}^{s,t}_{BP_*BP}(BP_*, BP_*X) \to H_{\text{cont.}}^s(\mathbb G_2;E_t(X))$$ where $X$ is a finite spectrum and $\mathbb G_2$ is $\mathbb S_2 \rtimes C_2$ with $\mathbb S_2$ the second Morava stablizer group at the prime $3$ and $C_2$ the cyclic group of order 2 acting on $\mathbb S_2$ via a (the?) lift of the Galois action. Furthermore, $E$ is the spectrum coming from lifts of the Honda formal group law (see page 127-128 for more details).
I am thinking we could write out an explicit description of this map in terms of the cobar complex and the usual resolution which computes continuous group cohomology. Indeed I think this should be straightforward to write down but I'm finding the literature rather impenetrable and the papers by Devinatz, Hopkins etc. to be rather abstract.