Restarting and Unstable - iMac 2010 and Error Code “2106F”

Sometimes that error message refers to some interruption or instability in the internet connection. Are you using a hardwired ethernet connection -- those are more stable than WiFi.

Also, see this article: Computers that can be upgraded to use OS X Internet Recovery - Apple Support

and this:

About macOS Recovery on Intel-based Mac computers - Apple Support

Some of the 2010 iMacs can use internet recovery but they need a firmware update first before that will work. Do you know if that was done to this Mac ever? If it was not, then internet recovery simply will not work. You can still install from the original install disks, those were either 10.6.3 or 10.6.4 if your iMac is really a 2010 model. You might be able to access those via eBay or by contacting an Apple Authorized Service Provider (third party Mac repair places that are authorized by Apple for various Mac services and repairs).

Have you tried normal built-in Recovery Mode (Command-R) instead of internet recovery? I had a 2008 iMac I used happily until recently (its power supply failed, worn out) and I could boot into normal Recovery and erase and install from the built in Recovery partition, internet recovery was not needed. I also have a 2010 MacBook Air and the Recovery mode internal recovery partition can be used to erase and install a new MacOS.

More advice from Apple for when internet recovery fails:

If you can't start up from macOS Recovery

If your Mac can't start up from its built-in macOS Recovery system, it might try to start up from macOS Recovery over the Internet. When that happens, you see a spinning globe instead of an Apple logo during startup: 

To manually start up from Internet Recovery, press and hold either of these key combinations at startup:

• Option-Command-R
• Shift-Option-Command-R
• Learn more

If startup from Internet Recovery is unsuccessful, you see a globe with a warning symbol (exclamation point):

If you see the globe and warning symbol, try these solutions:

• Make sure that your Mac can connect to the internet. If you're not prompted to choose a Wi-Fi network during startup, choose a network from the Wi-Fi menu , if available.
• Press Command-R at startup to try using the built-in Recovery system instead of internet Recovery.
• Connect to the internet using Ethernet instead of Wi-Fi, or vice versa.
• Connect to the internet from a different Wi-Fi or Ethernet network. Your network configuration might not allow the internet access that macOS Recovery needs.
• Try again later, because the issue might be temporary.
• Start up from another disk or volume, if available, or use a bootable installer to reinstall macOS.

If you still need help, please contact Apple Support.

2 things, first pleaser remember a 2010 iMac is considered an antique in computer years. Parts are no longer available and it will not run current versions of Mac OS. Also, Apple does not publish error codes.

The second thing, a 2010 is too old to support Internet Recovery. I doubt the computer came with the original SSDs, so that means your only alternative is to find a genuine Apple Snow Leopard upgrade DVD. Apple stopped selling them years ago, you can still find them on Ebay but there are a lot of counterfeits. Ask the seller if it's a genuine Apple DVD, if they say any but yes then pass until you find one. That DVD is wha you will need to startup the computer. I suspect the HD (at the very least) has failed.

If you cannot afford to buy a new Mac, consider an Apple refurbished Mac or buy a used Mac from a reputable reseller such as Macsales.com. NEVER EVER buy from E-Bay, Craigslist, Marketplace, Nextdoor, etc! Normally these machines a NOT setup correctly for resale and more often than not you are only inheriting someone elses problem.

Good luck!

Read through this article: https://9to5mac.com/2011/07/20/what-is-os-x-lion-internet-recovery-and-how-does-it-work/

In particular, this note this section on the Internet Recovery you just went through, noting the part in bold at the end of this quoted section:

"Lion Internet Recovery first runs a quick test of your memory and hard drive before putting up a limited interface that lets you select your preferred wireless network. From there, a Lion Recovery HD image downloads from Apple’s servers (note: This is only the recovery partition image, not the entire Lion installation) and your Mac boots into it and presents you with the same utilities and functions as Lion Recovery preloaded on the built-in recovery partition. Regardless of whether you booted into Lion Recovery from the recovery partition, by holding Command-R at startup, or your machine automatically downloaded Lion Recovery from Apple’s servers, this mode will let you re-download your copy of Lion from Apple’s servers (it’s a four gigabyte download) and begin the installation process. You will be prompted to enter the Apple ID and password you used to purchase Lion to authenticate your digitally signed copy of the operating system."

I am not familiar with Lion (have used MacOS earlier than Lion and also later than Lion) but it sounds like this Internet Recovery requires the previous owner's Apple ID. You might try to obtain some Apple Lion original install DVDs from various online sources like eBay, or perhaps consult with an Apple Authorized Service Provider, they often have installers for most any version of the MacOS. Or contact the person who sold you the Mac in this state?

I can't inspect your computer so I can't tell for sure that the above is what happened to you. Just in case the above is not in fact what happened to you, and you do have a complete installation of Lion successfully completed, you can try following the instructions to create a new administrator account at:

https://logi.wiki/index.php/Single_user_mode

If this does not work and in fact the complete Lion install requires the previous owner's Apple ID and password, maybe others here can help you out, I don't know how to get past this -- or consult an Apple Authorized Service Provider. The previous owner may have done you a disservice by not properly preparing the computer for sale, e.g. by wiping the disk and wiping off the old Apple ID.