@Gaea, although I do not have the Poco F3 in my hands myself, I suspect that especially the installation of /e/OS ‘S’ is the same as on a OnePlus 8T (see here).
If I were to install /e/OS ‘S’ (Android 12) on a Xiaomi POCO F3 (alioth), I would proceed as follows:
Before following these instructions please ensure that the device is currently using the Oneplus Oxygen firmware version mentioned below.
POCO F3 EEA (Euorpe) V22.214.171.124.SKHEUXM
POCO F3 Global V126.96.36.199.SKHMIXM
Unlock Pocco F3 bootloader (if not already done)
Flashing the vendor boot partition
Download this vendor_boot.img file
Power off your device, and boot it into bootloader mode:
With the device powered off, hold VOL(-) and POWER. Keep holding both buttons until the word “FASTBOOT” appears on the screen, then release.
Flash the downloaded vendor_boot.img to your device by typing
fastboot flash vendor_boot vendor_boot.img
Booting the eRecovery using fastboot
Flash the downloaded eRecovery to your device by typing
fastboot flash boot recovery-e-1.5.1-s-20221102231515-dev-alioth.img
Now reboot into eRecovery to verify the eRecovery installation.
Now iInstalling /e/OS ‘S’ from eRecovery
. . .
adb sideload e-1.5.1-s-20221102231515-dev-alioth.zip
Good luck and happy flashing!