Oh, so End(E_n) isnt known? That's surprising...

Ayala-Francis-Tanaka conjectured that Top(n) -> Aut(E_n) is an equivalence if and only if n<4. This is known for n<3, but as far as I am aware, even the \pi_0 statement is open in the other cases.

Thanks @ClarkBarwick, @DylanWilson. I kind of remember the Dwyer-Hess paper, but didn't remember it had anything to do with computing maps between operads. @archipelago I now remember asking my question several times after talks byvarious elements of {Ayala, Francis, Tanaka}!

I think there are maps $\mathrm{Top}(n) \to \mathrm{Aut}(E_n) \to hAut_\ast(S^n)$, factoring the obvious map. The first one is the "obvious" action on the little disks operad, while I think the second one is because a group acting on E_n gives rise to an action on the n-fold loop functor $(\mathcal{S}_*)_{\geq n} \to \mathcal{S}_*$.

Is that right?

@CharlesRezk I haven't seen the map Aut(E_n) -> hAut_*(S^n)$ in the literature, but this sounds right to me. The obvious map Top(n) -> hAut_*(S^n) factors through G(n)=hAut(S^{n-1}). I wonder whether Aut(E_n) -> hAut_*(S^n) does.

I extrapolate the existence of that map from Construction 3.7 in arxiv.org/abs/1707.03413 (which is doing it for n=1, if I understand it correctly.)