  We use the notion of isomorphism between two invariant vector fields to shed
new light on the issue of linearization of an invariant vector field near a
relative equilibrium. We argue that the notion is useful in understanding the
passage from the space of invariant vector fields in a tube around a group
orbit to the space invariant vector fields on a slice to the orbit. The notion
comes from Hepworth's study of vector fields on stacks.
