Adams spectral sequences for non-vector-bundle Thom spectra

When $R$ is one of the spectra $ku$, $ko$, $tmf$, ${MTSpin}^c$, $MTSpin$, or $MTString$, there is a standard approach to computing twisted $R$-homology groups of a space $X$ with the Adams spectral sequence, by using a change-of-rings isomorphism to simplify the $E_2$-page. This approach requires the assumption that the twist comes from a vector bundle, i.e. the twist map $X\to B{\mathrm{G}\mathrm{L}}_1(R)$ factors through $B\mathrm{O}$. We show this assumption is unnecessary by working with Baker-Lazarev’s Adams spectral sequence of $R$-modules and computing its $E_2$-page for a large class of twists of these spectra. We then work through two example computations motivated by anomaly cancellation for supergravity theories.