Robert Bryant's answer to Isometric embedding of SO(3) into an euclidean space mentions that there is an isometric embedding of the round tetrahedral space $ SO_3/A_4 $ into the round sphere $ S^6 $.

I like that fact. Dreaming along those lines, here are some more facts: (and I am interested in any similar examples that MO knows about)

-$S^4$ contains isometrically embedded copies of the the round projective plane $ \mathbb{R}P^2 $ and the round prism manifold $ S^3/Q_8 $ (this happens to be the manifold of complete flags in $ \mathbb{R}^3 $) where $ Q_8 $ is the eight element quaternion group

-This doesn't quite fit the theme of roundness but $ S^5 $ contains a weird distorted (not round but still Riemannian homogeneous, in particular with isometry group $ SU_2 $) copy of $ \mathbb{R}P^3 $ . This can be seen as the unit tangent bundle of the sphere $ S^2 $ inside $ T(\mathbb{R}^3)=\mathbb{R}^6 $. This weird $ \mathbb{R}P^3 $ is isometrically double covered by the Berger sphere $ S^3 $.

-$ S^6 $ contains an isometrically embedded round tetrahedral space $ SO_3/A_4 $ as previously mentioned

-$ S^8 $ contains a round $ \mathbb{R}P^3 $

-$ S^N $ contains a round projective space $ \mathbb{R}P^n $ where $ N=\frac{(n+1)(n+2)}{2}-2 $, moreover $ N $ is the minimal such dimension.

Moreover all these embeddings are equivariant with respect to the action of the Isometry group (which is transitive-- these manifolds are all Riemannian homogeneous in addition to being round).

I am aware that for sufficiently large $ N $ every Riemannian manifold isometrically embeds in round $ S^N $. What I am really looking for are **(equivariant) isometric embeddings of round (homogeneous) Riemannian manifolds into low dimensional round spheres.** The properties in parentheses are especially interesting.

Disclaimer: ~everything is smooth I know Nash-Kuiper does very low dimensions for $ C^1 $~