Algebraic groups of homotopy classes of automorphisms and operadic Koszul duality

Given a simply connected space $X$, there are several, a priori different, algebraic groups whose groups of $\mathbb{Q}$-points are isomorphic to the group of homotopy classes of homotopy automorphisms of the rationalization of $X$. We will show that two of these different algebraic groups are isomorphic using the theory of operadic Koszul duality. As a by-product of the techniques we use, we deduce some isomorphisms of sets of homotopy classes of maps that might be viewed as Eckmann-Hilton dual to some well-known isomorphisms.