Relatively finite measure-preserving extensions and lifting multipliers by Rokhlin cocycles

We show that under some natural ergodicity assumptions extensions given by Rokhlin cocycles lift the multiplier property if the associated locally compact group extension has only countably many L^\infty-eigenvalues. We make use of some analogs of basic results from the theory of finite-rank modules associated to an extension of measure-preserving systems in the setting of a non-singular base.