The OS installation asks for an ID and writes a connection failure

Try booting using Command + Option + R instead which should not require you to authenticate with an AppleID or it will access Internet Recovery Mode for a more recent macOS online installer if the Mac supports a newer OS.

FYI, it is always a good idea to provide the exact model of your Mac when asking for assistance online since different models may have different options available. You can get the exact model by entering the system serial number here:

Check Your Service and Support Coverage - Apple Support

You may have to have Apple reinstall it, Lots of similar complaints, see if this works...

 Start up from macOS Recovery

Option-⌘-R

Upgrade to the latest macOS compatible with your Mac.

In Disk Utility>View, select Show all Devices, highlight the top left entry.

APFS won't work for Sierra or earlier, Sierra or earlier needs MacOS Extended Journaled. High Sierra won't work for APFS on earlier Macs.

What has often worked for others is to use this Restore Method instead of CMD+r keys...

Option-⌘-R

Upgrade to the latest macOS compatible with your Mac.

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