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finally had a closer look at Gómez-Tato, Halperin & Tanré 2000, but I don’t see clearly yet on the key point:
As opposed to Brown & Szczarba 1995 these authors develop a theory that does not have explicit $\pi_1$-actions on dg-algebras, but instead is formulated in terms of dg-algebra valued functors on the category of elements of a simplicial set modelling $B \pi_1$. On the other hand, the examples offered towards the end (Examples. 6.6 - 6.8) suddenly speak of $\pi_1$-actions via dg-algebra automorphisms, but seemingly not about those functors on categories of elements that occupy the bulk of the article.
I am getting the impression that the connective tissue between theory and examples is meant to be Prop. 3.17, which claims, it seems (without further proof or argument), that from the previous Theorem 3.12 one dg-algebra automorphism may be extracted. Currently I have trouble even parsing the ingredients appearing in Prop. 3.17. E.g. the “$\phi$” here seems to be different from the $\phi$ on the previous pages and throughout.
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