Red quasi-stellar objects (QSOs) are a subset of the quasar population with colours consistent with reddening due to intervening dust. Recent work has demonstrated that red QSOs show special radio properties that fundamentally distinguish them from normal blue QSOs, specifically a higher incidence of low-power radio emission (1.4 GHz luminosities L$_{rm 1.4} approx 10^{25}$ - $10^{27}$ W Hz$^{-1}$) that is physically compact when imaged by arcsecond-resolution radio surveys such as FIRST. In this work, we present e-MERLIN imaging of a set of intermediate-redshift ($1.0<z<1.55$), luminous (bolometric luminosities L$_{bol} approx 10^{46}$ - $10^{47}$ erg s$^{-1}$) red and normal QSOs carefully selected to have radio properties that span the range over which red QSOs show the most divergence from the general population. With an angular resolution $times25$ better than FIRST, we resolve structures within the host galaxies of these QSOs ($> 2$ kpc). We report a statistically significant difference in the incidence of extended kpc-scale emission in red QSOs. From an analysis of the radio size distributions of the sample, we find that the excess radio emission in red QSOs can be attributed to structures that are confined to galaxy scales ($< 10$ kpc), while we confirm previous results that red and normal QSOs have similar incidences of radio jets and lobes on circumgalactic or larger scales ($> 10$ kpc). Our results indicate that the primary mechanism that generates the enhanced radio emission in red QSOs is not directly connected with the nuclear engine or accretion disc, but is likely to arise from extended components such as AGN-driven jets or winds.
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