Conformal Killing 2-forms on 4-dimensional manifolds

We study 4-dimensional simply connected Lie groups $G$ with left-invariant Riemannian metric $g$ admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action, or the metric is half conformally flat. In the first case, the problem reduces to the study of invariant conformally K\"ahler structures, whereas in the second case, the Lie algebra of $G$ belongs (up to homothety) to a finite list of families of metric Lie algebras.