Sure. I was specializing a more general result. Let $C$ be a coalgebra, $A$ an algebra. $Hom_R(C,A)$ is an $R$-algebra. Let $G(C,A)$ be the submonoid of those $f$ such that $f\eta = \eta$ and $\epsilon f = \epsilon$. One looks for sufficient conditions on $C$ and $A$ for when $$G(C,A)$ is a group. Then one must assume $C_0$ is $R$-free, since the group hypothesis refers to $A$.

Chin up, push on, Tara. Ignore the most formidable parts: the parts are written so as to be independently readable. The more formidable parts are usually there to fill gaps in the literature, so precisely the parts that are hardest to find elsewhere. And thanks, Neil.

No, not there, but a spectral sequence of the sort requested does appear in ``The geometry of iterated loop spaces'', pages 155-156. That is probably the first reference to such a spectral sequence, but not the best. I believe Kraines and Lada made some calculational use of such a spectral sequence.