HA updates - update the Pi?

Apologies for the likely dumb question.

I downloaded the HA image for * Raspberry Pi 4 Model B (1 GB, 2 GB, 4 GB and 8 GB model) 64-bit(64-bit is required for 8 GB model)

and installed that an RPi. I update HA when HA says there is an update, but do I need to update the underlying RPi itself? I can’t wrap my head about how HA is installed on the Pi. For instance with PiHole, I update the PiHole via PiHole -up but also have to VNC into the PI occasionally and update the Pi OS.

If I need to update the underlying OS, how do I do that? The recent sudo CVE is what brought this to mind.

Thanks.

If you use an image then Home Assistant OS is the base OS. I think You should be able to update in the same place as supervisor and core, I even think it updates automatically.

Only the supervisor updates automatically (after a certain unknown time). You have to manually update the OS in the place mentioned.