Install Big Sur on external SSD

I am running Mojave. I attempted to install Big Sur to my Mojave system disk, but one of the reboots hung. Restarting hung as well.
I restored my Mojave system disk and am now trying to install Big Sur on an external disk. I run the “Install macOS Big Sur app” and point it to the external disk. It runs and installs the folder “macOS Install Data” on the external disk.When it asks to reboot, I hold the Option key. However, the external disk is not listed.
Do you have any suggestions?

Have you tried using a thumb drive to create a bootable installer and then installing from that onto the external drive?

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You shouldn’t need to hold the option key when asked to reboot, as it should reboot by itself on the external drive where the OS was installed, doesnt’it?

No, it does not.

After running the install app, open System Preferences and click on Startup Disk. Your external disk should be one of the choices. Click on it, close system preferences, and reboot.

If you don’t see it as one of the choices, I would reformat the external drive and retry the install. You may want to re-down the install application if you continue to have issues.

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I have had no luck installing Big Sur on an empty external disk.
However, after restoring my Majove system disk, I was able to install Big Sur onto it.
Thanks for the suggestions.