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I think your problem is not constrained enough to have an interesting answer. Notice that your intertwining condition can be rephrased by saying that $g: B \to A$ is a homomorphism of $R$-algebras, where $A$ is given the structure of $R$-algebra given by $f$. In these terms, what you are looking for is the comma category $\mathcal{C} = (\mathbf{Alg}_R \downarrow A)$, whose objects are precisely the pairs $(B, g)$ as above, and whose morphisms $\operatorname{Hom}_{\mathcal{C}}((B, g), (B', g'))$ are the $R$-algebra homomorphisms $h: B \to B'$ such that $g = g' \circ h$. I am not sure it is possible to capture this beast with a cohomology group of any sort.