If I have a fibration, perhaps with twisting data respecting the fibration, is there a Serre spectral sequence computing cobordism of the total space?

An example that I'm particularly interested in is twisted spin cobordism for the fibration of classifying spaces of the groups $\mathbb{Z}/2 \to \mathbb{Z}/4 \to \mathbb{Z}/2$ with twisting bundle given by the sign representation on the base, its pullback on the total space, and the trivial twisting bundle on the fiber.

By twisted spin cobordism I mean I want to consider (unoriented) manifolds with a map to the space along with a spin structure on the tangent bundle direct sum the pullback of the twisting bundle.