Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. See for example
Does there exists a compact connected manifold $ M $ and two metrics $ g_1,g_2 $ such that $$ dim(Iso(M,g_1))=N(M)=dim(Iso(M,g_2)) $$ but the isometry groups $ Iso(M,g_1) $ and $ Iso(M,g_2) $ are non isomorphic Lie groups?
I'm mostly interesting in the case that $ Iso(M,g_1) $ and $ Iso(M,g_2) $ are semisimple or even simple. I'm also mostly interested in the connected component of the identity. So examples like the flat metric on the hexagonal torus $ T^2 $ versus the square torus $ T^2 $ don't really concern me because the isometry groups agree on the identity component.
What I'm really looking for would be something like $ N(M)=21 $ and $ Iso(M,g_1)=Sp(3) $ while $ Iso(M,g_2)=Spin_7 $
This question
Uniqueness of metric from compact simple group of isometries
gives examples like round $ S^5=SO_6/SO_5 $ and Berger $ S^5=SU_3/SU_2 $. But the Berger sphere does not have an isometry group of maximum dimension $ dim(Iso(SU_3/SU_2))=8 \neq 15= dim(Iso(SU_3/SU_2)) = N(S^5) $