Let $ M $ be the Moebius band. In other words, the total space of the nontrivial line bundle over the circle. Can we equip $ M $ with a metric such the the isometry group acts transitively?
My intuition is that yes we can because $ M $ is the total space of a vector bundle over a compact Riemannian homogenous space (the circle).
However thanks to Ben McKay for the argument that $ M $ does not have a transitive isometry group. Also this fact is stated in "Isometries of 2-Dimensional Riemannian Manifolds into Themselves" by Sumner Byron Myers.
More generally let $ E $ be the total space of a vector bundle with base space $ B $ where $ B $ is a compact Riemannian homogeneous manifold. When does there exist a metric on $ E $ such that $ E $ is Riemannian homogeneous? Does the bundle have to be equivariant with respect to the action of the isometry group of $ B $?