Recovery mode error 5101F

What is the exact model of your Mac?

What version of macOS is currently installed?

What version of the macOS installer are you accessing in Internet Recovery Mode?

Try creating a bootable macOS USB installer:

https://support.apple.com/en-us/HT201372

I had an issue in booting the Big Sur installer from Internet Recovery Mode yesterday and I think I had that error number (unless I'm transposing the numbers and it was -15xx). But the laptop I was using was being managed by our organization so I'm thinking it may have been configured to block Big Sur although another tech said it should have worked since the MDM is only set to block Big Sur installs from System Preferences and the App Store.

After they unlocked the laptop for me I still had issues installing Big Sur using the USB installer (I didn't try Internet Recovery Mode since it took so long to boot & download). Took me another whole day to get Big Sur installed.

Make sure you have a good backup before attempting the upgrade to Big Sur. I suggest you create a bootable macOS USB installer so that you don't have to rely on a very slow download of Big Sur each time you attempt the install. Trust me it has been painful even when using the USB installer (it took an hour just to fail, then another hour once I was able to get it to "resume" the installation from where is left off at).

In Internet Recovery, these low negative thousands error messages indicate that your provided Wi-Fi Network is inadequate in some fashion, and cannot be used (for this purpose) in its current state.

Common problems include:

Hidden Network-name

required use of a proxy server to get Internet access

required use of a login page to get Internet access

Use of a login server or certificate to get Internet access

Use of PPPoE in the Mac to get Internet access (typically only applies to DSL)

Use of Fixed IP address rather than good old DHCP

Variance of the time by more than five minutes

Failure of Router to provide workable DNS server addresses, or providing 0.0.0.0 [may produce -2002f]

If you have an Ethernet port, you may in some cases be able to move your Mac close to the Router and connect to the Router using Ethernet.

The higher the number, the more obscure the problem.

If your battery was completely depleted, the Date&Time could have reverted to the default, which will not work. There is a terminal command to reset to current Internet Date&Time -- post back for assistance.

JupiterSpaceport wrote:

...

I also wonder how its getting access if I never gave it the password?

It CAN"T Get access if you never, ever provided the password. But if you previously provided the Wi-Fi password, it could have saved that password in parameter RAM.

This is a defect in YOUR Network. This is not a defect in the computer, and there is no point returning it if you are not going to fix your network.