Let $M$ be a connected and second-countable manifold, and $d,d'$ be $2$ complete compatible metrics on it. The isometry group of $(M,d)$, denoted as $\mathrm{Iso}(d)$, is equipped with the compact-open topology.

We say that $\mathrm{Iso}(d')$ is a homotopic subgroup of $\mathrm{Iso}(d)$, if there is a topological group embedding $\sigma : \mathrm{Iso}(d') \rightarrow \mathrm{Iso}(d)$ such that $\forall g \in \mathrm{Iso}(d')$, $g$ is homotopic to $\sigma(g)$ as self maps of $M$.

If $\mathrm{Iso}(d)$ is also a homotopic subgroup of $\mathrm{Iso}(d')$, then we say that $(M,d)$ and $(M,d')$ achieve the same symmetry. If $\mathrm{Iso}(d)$ is never a strict homotopic subgroup of $\mathrm{Iso}(d')$ for any $d'$, then we say that $(M,d)$ achieves a maximal symmetry.

Q$1$: Do homogeneous spaces, equipped with an arbitrary invariant smooth metric, always achieve maximal symmetries? A previous post says that the homogeneous flat metric on $\Bbb{R}^n$ achieves the unique maximal symmetry among normable metrics. However maximal symmetry may not be unique, e.g. $\Bbb{R}^n$ is diffeomorphic to $\Bbb{H}^n$ but their symmetries are incomparable. Edit: This is answered negatively by Robert Bryant.

Q$2$: Can maximal symmetries on compact smoothable manifolds always be realized smoothly (i.e. if $(M,d)$ achieves a maximal symmetry, then there exists a Riemann structure $(M,g)$ achieving the same symmetry)?