JWST IFU observations uncover host galaxy continua in extremely red and obscured quasars

Uncovering bright quasars' host galaxies at cosmic noon is challenging because of the high contrast between the quasar and its host and redshifted light, making them primarily visible in the infrared. We present JWST NIRSpec integral field unit (IFU) observations of six extremely red quasars (ERQs) at $z=2.4-2.9$ and two dust-obscured quasars at lower redshifts. Using image decomposition across the spectral range, we successfully separate quasar and host galaxy continuum emission, model host morphologies, and extract spectra. The ERQs and obscured quasars have compact host galaxies with half-light radii of 1.4$-$2.9 kpc and stellar masses of 10$^{10.6-10.9}$ $M_{\odot}$. Their stellar masses are consistent with the average stellar mass of quasar hosts as expected from abundance matching and clustering analysis. Most of the quasars in our sample exhibit significant spatial offsets (0.4$-$1.3 kpc) between the quasar and host galaxy, potentially caused by post-merger dynamics or non-uniform dust obscuration. The ERQs reside 0.5$-$2 dex above the local black hole-stellar mass relation, similar to other heavily obscured populations such as HotDOGs, optically selected quasars at cosmic noon, and high-redshift SMBH candidates identified with JWST. However, this "over-massive" feature might be attributed to selection bias. Compared to HST-based studies, our JWST measurements reveal more compact host galaxies, smaller Sersic indices, and lower stellar masses, likely because of improved resolution, more accurate modeling, and minimal line contamination. These findings highlight the unique capabilities of JWST IFU in revealing quasar host galaxy properties and potential evolutionary stages of obscured quasars at cosmic noon.