I'm having an old MacBook Air 4.2, (mid 2011, SSD, 4 Gb memory), which have been running Catalina for quite some time despite that the max OS is High Sierra. I have been using the great work by dosedudes1 and his Catalina patcher.

Unfortunately he didn't continue with patches for newer OS:s, but have helped with a new patcher. This patcher even handles Macs with graphic cards from 2008 - 2011.
Using the patcher i got the mentioned Mac up and running with Big Sur after just over an hour.

How to install Big Sur on unsupported Mac To install Big Sur on an unsupported Mac, you must first create a bootable USB drive using a special patcher app. Then you need to use the USB drive to install Big Sur on the unsupported Mac, and, lastly, you must install the needed patches. Creating a bootable USB stick using the Big Sur patcher. When Not To Update an Unsupported Mac To Big Sur. Just because you can install Big Sur doesn’t necessarily mean that you should. First, it’s a very complicated process as the Big Sur update is one of the most significant in recent times, making it more difficult for patches to be created. Jun 27, 2020 macOS Big Sur. First thing first, the version number. Yes, we are now looking at macOS 11, after roughly 20 years of macOS 10/Mac OS X incremental updates. Well, with the release of this new version which will be available on Apple Silicon-based Macs and which introduces a totally new design, Apple taught it was time. Unsupported Macs supported by ‘Patched Sur’ First of all, make sure that your Mac is not in the official macOS Big Sur list of compatible Macs.At the time of writing, the following Macs are.

Fantastic !!!
Mac os big sur unsupported mac github
Patcher manual
The manual for the OpenCore Legacy Patcher is nice and there are also a video that will help you for the whole process.
Create bootable USB

You start with creating a bootable USB with info from this post. Just pick Big Sur instead of the mentioned High Sierra.


Mac

Building/installing the patcher

After that follow these instructions,on how to install and use the OpenCore Legacy patcher.

Good luck !

Remarks

Mac Os Big Sur Unsupported Mac Github

Sur

Mac Os Big Sur Public Beta Unsupported Mac

  • Erase/format the MAC HD/SSD before installation !! I really think that this does it more simple for the installer !!
    • 'Mac OS Extended(journaled)' with an old non SSD HD
    • With en SSD use 'APFS'
  • The Mac rebooted several times during install.
  • Don't remove the USB until at the very end.
  • If you, like me, have a Mac which need the legacy video acceleration support please note that the Mac have a painfully slow display update before you have applied the 'Post-Install volume Patch'.
  • Updating the Mac from Big Sur 11.4 to 11.5, made the messages.app stop working fixed in release 0.2.4
  • Please note that 'Nightly Builds' are available but are for testing purposes only !!!