The so-called Iterated Fission Probability (IFP) method has provided a major breakthrough for the calculation of the adjoint flux and more generally of adjoint-weighted scores in Monte Carlo eigenvalue calculations. So far, IFP has been exclusively devoted to the analysis of the standard $k$-eigenvalue equation, by resorting to a formal identification between the adjoint fundamental eigenmode $\varphi ^\dagger_k$ and the neutron importance $I_k$. In this work, we extend the IFP method to the $\alpha$-eigenvalue equation, enabling the calculation of the adjoint fundamental eigenmode $\varphi ^\dagger_k$ and the associated adjoint-weighted scores, including kinetics parameters. Such generalized IFP method is first verified in a simple two-group infinite medium transport problem, which admits analytical solutions. Then, $\alpha$-adjoint-weighted kinetics parameters are computed for a few reactor configurations by resorting to the Monte Carlo code Tripoli-4®, and compared to the $k$-adjoint-weighted kinetics parameters obtained by the standard IFP. The algorithms that we have developed might be of interest in the interpretation of reactivity measurements, in the context of reactor period calculations by Monte Carlo simulation.