I am trying to install Norton Utilities Ultimate and I get a message that System Checkup 3.5 3.5.4.12 needs to be uninstalled. When I go to System Checkup 3.5 the Modify and Uninstall buttons are grayed out, so I cannot uninstall it.

How Do I Uninstall SCU (System Checkup)
If you need assistance with uninstalling any iolo products, please contact customer care through our Customer Care Portal and agents will be happy to assist you.
https://support.iolo.com/support/solutions/articles/44001697739

R U Windows Administrator user account?

Note: I do not run System Checker nor Norton Utilities Ultimate 

Maybe, try delete systemcheckup.exe, then try uninstall System Checkup from your computer using its uninstaller located at:

C:\Program Files (x86)\iolo\System Checkup\uninstscu.exe /uninstall

C:\Program Files\iolo\System Checkup\uninstscu.exe /uninstall

source Google

Maybe, Revo Uninstaller Free