You probably don't have a real solution. Since they do not provide older binaries for GitHub for Windows, you cannot install them if you do not have the installation file.
GitHub recently combined GitHub for Windows and GitHub for Mac into one application: GitHub Desktop and requires Windows 7 or later :
https://desktop.github.com/
In short, you cannot use any official GitHub desktop application, as they no longer support XP.
Alternatively, you can use GitHub in your browser, push commits from the command line or desktop client that supports XP.
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