Commutativity preserving transformations on conjugacy classes of finite rank self-adjoint operators

Let H be a complex Hilbert space and let C be a conjugacy class of rank k self-adjoint operators on H with respect to the action of the group of unitary operators. Under the assumption that dim⁡H≥4k we describe all bijective transformations of C preserving the commutativity in both directions. In particular, it follows from this description that every such transformation is induced by a unitary or anti-unitary operator only in the case when for every operator from C the dimensions of eigenspaces are mutually distinct.