Question: Let $ M $ be a manifold. Let $ g $ be a metric on $ M $ . Let $ K_g $ be the connected component of the identity in the maximal compact subgroup of the isometry group $ Iso(M,g) $. Does there exists a unique compact connected group $ K_M $ such that, for every metric $ g $ on $ M $, the group $ K_g $ is isomorphic to a subgroup of $ K_M $? The final restriction I wish to place on $ K_M $ is that it cannot be chosen to be arbitrarily large. In particular, there must exist at least one metric $ g^* $ on $ M $ such that $ K_{g^*} \cong K_M $.
Here are some examples:
$ M $, $ K_M $ (a metric $ g^* $ such that $ K_{g^*}=K_M $)
The two sphere $ S^2 $, $ K_{S^2}=SO_3(\mathbb{R}) $ (round metric)
The projective plane $ \mathbb{R}P^2 $, $ K_{\mathbb{R}P^2}=SO_3(\mathbb{R}) $ (round metric)
The two torus $ T^2 $, $ K_{T^2}=SO_2(\mathbb{R}) \times SO_2(\mathbb{R}) $ (flat metric)
The plane, $ K_{\mathbb{R}^2}=SO_2(\mathbb{R}) $ (flat or hyperbolic metric)
The surfaces of genus $ g \geq 2 $, the trivial group (any metric)
I am primarily interested in the case of compact connected manifolds that admit a metric with respect to which the isometry group is homogeneous (i.e. Riemannian homogeneous). For example the sphere, projective plane, and torus.
A negative answer to my question would take the following form: a manifold $ M $ and two metrics $ g_1,g_2 $ on $ M $ such that neither $ K_{g_1} $ nor $ K_{g_2} $ embeds in the other. Here $ K_g $ denotes the connected component of the maximal compact subgroup of the isometry group $ Iso(M,g) $
Certainly such non comparable groups of isometries exists for the same manifold if we allow the group of isometries to be non connected (the square and hexagonal tori) and if we allow the group of isometries to be noncompact (the flat and hyperbolic metrics on the plane). See "Maximal symmetry" metric for a manifold?. However I am restricting the the discussion to the case of groups of isometries which are both compact and connected.