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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.

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It might not be quite of the form that you're looking for, but for a fibre sequence $F \to E \to B$, there is an Atiyah-Hirzebruch-Serre spectral sequence of the form

$$H^*(B; R^*(F)) \implies R^*E$$

for a generalized cohomology theory $R$. Here $H^*(B; R^*(F))$ is the singular cohomology of $B$ with coefficients in the local system $R^*(F)$. In the case that you mention, you get a group cohomology calculation

$$H^*(\mathbb{Z} / 2; R^*(B\mathbb{Z} / 2)) \implies R^*(B\mathbb{Z} / 4).$$

I'm not sure I totally understand what you mean by "twisting bundle" here, but if you mean that you want to modify this by some sort of local system (or Thom isomorphism) $L$ on the base (and its pullback to the total space), that works, too:

$$H^*(\mathbb{Z} / 2; R^*(B\mathbb{Z} / 2; L)) \implies R^*(B\mathbb{Z} / 4; p^* L).$$

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