Re running the 12.0.1 install and, on the last reboot, ending up still on Big Sur, this happened to me a couple of times. The only solution I found was as follows:
BUT FIRST, you'd better have a couple of current backups handy from which to restore your data. If not, reboot the target BS volume and make a backup. I use TM, and also migration to an external SSD
Now boot to an external USB or external SSD, and erase the volume in question using Disk Utility, recreating it as an APFS volume (the default).
Then do a clean install. I do it from a usb or external SSD session. The Installer will ask which volume you want to install it onto, pick the empty one you just recreated. Last reboot will prompt you for Data Migration. As a data source I've successfully used TM and also the external usb>SSD drive.
There may be other ways, that's just what I would do. Installing "over the top" always seems higher risk to me, and on more than a few attempts it hasn't worked.