Loday constructions on twisted products and on tori

We develop a spectral sequence for the homotopy groups of Loday constructions with respect to twisted cartesian products in the case where the group involved is discrete. We show that for commutative Hopf algebra spectra Loday constructions are stable, generalizing a result by Berest, Ramadoss and Yeung, but prove that several Loday constructions of truncated polynomial rings with reduced coefficients are not stable by investigating their torus homology.