For a 1-connected space $X$ with the homotopy type of a finite CW-complex, the homotopy automorphisms $\pi_0\,\mathrm{hAut}(X)$ are known to have the structure of an arithmetic group up to finite kernel. In fact, there are at least two such constructions in the literature:

• Sullivan's in Theorem 10.3 of Infinitesimal computations in topology.
• Wilkerson's in Theorem B of Applications of minimal simplicial groups.

In both cases the arithmetic group structure is obtained from the inclusion (up to finite kernel) of $\pi_0\,\mathrm{hAut}(X)$ into the homotopy automorphisms $\pi_0\,\mathrm{hAut}(X_\mathbb{Q})$ of the rationalisation $X_\mathbb{Q}$. Both references construct an $\mathbb{Q}$-algebraic group structure on $\pi_0\,\mathrm{hAut}(X_\mathbb{Q})$, but since their constructions proceed along different lines it is not clear these coincide.

Are Sullivan's and Wilkerson's algebraic group structures on $\pi_0\,\mathrm{hAut}(X_\mathbb{Q})$ the same?

For a 1-connected space $X$ with the homotopy type of a finite CW-complex, the homotopy automorphisms $\pi_0\,\mathrm{hAut}(X)$ are known to have the structure of an arithmetic group up to finite kernel. In fact, there are at least two such constructions in the literature:

• Sullivan's in Theorem 10.3 of Infinitesimal computations in topology.
• Wilkerson's in Theorem B of Applications of minimal simplicial groups.

In both cases the arithmetic group structure is obtained from the inclusion (up to finite kernel) of $\pi_0\,\mathrm{hAut}(X)$ into the homotopy automorphisms $\pi_0\,\mathrm{hAut}(X_\mathbb{Q})$ of the rationalisation $X_\mathbb{Q}$. Both references construct an $\mathbb{Q}$-algebraic group structure on $\pi_0\,\mathrm{hAut}(X_\mathbb{Q})$, but since their constructions proceed along different lines it is not clear these coincide.

Are Sullivan's and Wilkerson's algebraic group structures on $\pi_0\,\mathrm{hAut}(X_\mathbb{Q})$ the same?