  We use photometric redshifts and statistical background subtraction to
measure stellar mass functions in galaxy group-mass
($4.5-8\times10^{13}~\mathrm{M}_\odot$) haloes at $1<z<1.5$. Groups are
selected from COSMOS and SXDF, based on X-ray imaging and sparse spectroscopy.
Stellar mass ($M_{\mathrm{stellar}}$) functions are computed for quiescent and
star-forming galaxies separately, based on their rest-frame $UVJ$ colours. From
these we compute the quiescent fraction and quiescent fraction excess (QFE)
relative to the field as a function of $M_{\mathrm{stellar}}$. QFE increases
with $M_{\mathrm{stellar}}$, similar to more massive clusters at $1<z<1.5$.
This contrasts with the apparent separability of $M_{\mathrm{stellar}}$ and
environmental factors on galaxy quiescent fractions at $z\sim 0$. We then
compare our results with higher mass clusters at $1<z<1.5$ and lower redshifts.
We find a strong QFE dependence on halo mass at fixed $M_{\mathrm{stellar}}$;
well fit by a logarithmic slope of $\mathrm{d}(\mathrm{QFE})/\mathrm{d}\log
(M_{\mathrm{halo}}) \sim 0.24 \pm 0.04$ for all $M_{\mathrm{stellar}}$ and
redshift bins. This dependence is in remarkably good qualitative agreement with
the hydrodynamic simulation BAHAMAS, but contradicts the observed dependence of
QFE on $M_{\mathrm{stellar}}$. We interpret the results using two toy models:
one where a time delay until rapid (instantaneous) quenching begins upon
accretion to the main progenitor ("no pre-processing") and one where it starts
upon first becoming a satellite ("pre-processing"). Delay times appear to be
halo mass dependent, with a significantly stronger dependence required without
pre-processing. We conclude that our results support models in which
environmental quenching begins in low-mass ($<10^{14}M_\odot$) haloes at $z>1$.
