I know this is not a common scenario, but for the few people that upgraded a Mac originally equipped with a Core Duo processor to a Core 2 Duo processor, here is the procedure. I have an iMac version 4,1 that I upgraded years ago, and it worked. The person who found the solution had an upgraded Mac Mini. You will need a second computer that is Lion capable for this tip to work.

Mac: When Mountain Lion was released last summer it dropped support for older generation Macs from 2006-2008. There were workaround to get it to work, but they were far from user-friendly. Jul 30, 2012 Question: Q: Mountain Lion on an unsupported mac More Less Apple Footer This site contains user submitted content, comments and opinions and is for informational purposes only. Install Mountain Lion On Unsupported Mac In order to start fresh, you will need access to a functional computer running OS X Lion 10.7.5 or later. STEP 1: Download macOS Mojave The full operating system is a free download for anyone who has purchased Mac OS X Snow Leopard, Lion, or Mountain Lion or has a Mac preloaded with OS X Mavericks.

OS X Mavericks on Unsupported Macs. MacOS Extractor, OS X Patcher, and MacPostFactor are apps that guide you through patching and installing OS X El Capitan (10.11), Yosemite (10.10), Mavericks(10.9), or Mountain Lion (10.8) on your older Mac.

As always, Make sure you have a backup of the original install! This worked for me, but always be prepared for the worst.
I made, and used, a Lion Mac OS X Install ESD on a thumb drive. Instructions to make one can be found all over the Mac centric web sites. Then follow this procedure:

• Put the unsupported Mac in Target Disk mode and plug it into a Lion supported Mac.
• Run the Lion installer and make sure to choose your unsupported Mac's hard drive. Let the install finish.
• Boot back into the supported Mac's OS, and put the unsupported Mac back to target disk mode.
• Delete the PlatformSupport.plist at /System/Library/CoreServices on the unsupported Mac's hard drive.
• Reboot the Unsupported Mac and it should boot into Lion normally.

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High Sierra On Unsupported Mac

crarko adds:

Install Os X Mountain Lion On Old Unsupported Macs

I don't have the equipment to try this, but it seems plausible. As recommended in the hint, have a complete system backup handy in case something goes awry.]