Homotopy Type of Diffeomorphism Group of Lie Group

Question

Let $G$ be a finite dimensional connected Lie group and $Diffeo(G)$ be the diffeomorphism group of the underlying manifold. Is it true that $Diffeo(G)$ has the homotopy type of a finite dimensional Lie group? I can't seem to find a counterexample.

Answer 1

Torus of dimension $\ge 25$ is a counter example. See here.

Answer 2

A lower dimensional example is $S^1\times S^2$.

Hatcher calculated the homotopy type of $Diff(S^1\times S^2)$: It is the one of $O(2)\times O(3)\times \Omega SO(3)$. As the second homotopy group of this space does not vanish since $\pi_2(O(2)\times O(3)\times \Omega SO(3))\cong \mathbb{Z}$ holds, it cannot have the homotopy type of a finite dimensional Lie group.