  When $R$ is one of the spectra $\mathit{ku}$, $\mathit{ko}$, $\mathit{tmf}$,
$\mathit{MTSpin}^c$, $\mathit{MTSpin}$, or $\mathit{MTString}$, 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 $X\to B\mathrm{GL}_1(R)$ 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.
