In question Relations between Stiefel-Whitney classes the relations between Stiefel-Whitney classes on manifold are obtained.
My question is that do we have additional relations between Stiefel-Whitney classes on oriented mapping torus? If yes, what are they?
Also do we have relations between Stiefel-Whitney classes and Pontryagin classes (mod 2) on mapping torus?
(Mapping torus is a fiber bundle over $S^1$)