According to this question, Hansen proved that the space $\mathrm{Aut}_0(\mathbb{S}^2)$ of orientation preserving self-homotopy equivalences of the 2-sphere is homotopy equivalent to $\mathrm{SO}(3)\times\mathbf{\Omega}$, where $\mathbf{\Omega}$ is the universal cover of the connected component of $\Omega^2(\mathbb{S}^2)$ containing the generator of $\pi_2(\mathbb{S}^2)$ (or any other component, they are all homotopy equivalent).

Let’s now consider the space $\mathrm{Aut}^{\mathrm{pr}}_0(\mathbb{S}^2)$ of pairs $(f,H)$ where $f$ is a self-homotopy equivalence of the 2-sphere and $H$ is ~~an homotopy between f and the identity function~~ a connected component of the space of homotopies between $f$ and the identity function. You can see $H$ as a "proof" that $f$ is orientation preserving and $\mathrm{Aut}^{\mathrm{pr}}_0(\mathbb{S}^2)$ is the space of orientation preserving self-homotopy equivalences $f$ of the 2-sphere where you do not throw away the proof that $f$ is orientation preserving (the "pr" stands for "proof relevant").

**What is the homotopy type of $\mathrm{Aut}^{\mathrm{pr}}_0(\mathbb{S}^2)$?**

One can reasonably expect that the answer is $\mathrm{SO}(3)$, the noise in $\mathrm{Aut}_0(\mathbb{S}^2)$ looks very similar to the missing homotopy, but I don’t know how to prove it.

Also, thanks to Springer I can’t find an online version of Hansen’s paper, so maybe this is already in his paper.