We give a negative answer to a question put by Nadkarni: Let S be an ergodic, conservative and nonsingular automorphism on $(X̃{,}_{X̃},m)$. Consider the associated unitary operators on $L²(X̃{,}_{X̃},m)$ given by $U{̃}_{S}f=\surd (d\left(m\circ S\right)/dm)·\left(f\circ S\right)$ and $\phi ·U{̃}_S$, where φ is a cocycle of modulus one. Does spectral isomorphism of these two operators imply that φ is a coboundary? To answer it negatively, we give an example which arises from an infinite measure-preserving transformation with countable Lebesgue spectrum.