I don't think I've ever had success using the built in auto update functionality. (At least, not recently). To ensure I have the latest version I always use the "current" links. While you could argue the SHA2 won't run on SP1 or SP2, I have no reason to run anything older than SP3. For example, my HP all-in-one printer that I bought a few months ago requires a minimum of SP3 along with Internet Explorer 8.
https://fpdownload.macromedia.com/pub/flashplayer/latest/help/install_flash_player_ax.exe
https://fpdownload.macromedia.com/pub/flashplayer/latest/help/install_flash_player.exe
https://fpdownload.macromedia.com/pub/flashplayer/latest/help/install_flash_player_ppapi.exe
https://fpdownload.macromedia.com/get/flashplayer/current/support/uninstall_flash_player.exe
Windows XP will be supported straight through the end of Flash Player support next year, so there's no risk of downloading a new version that incompatible with Windows XP. Adobe itself said as much in its July 25, 2017 press release Flash & The Future of Interactive Content.
"Adobe will continue to support Flash on a number of major OSs and browsers that currently support Flash content through the planned EOL. This will include issuing regular security patches, maintaining OS and browser compatibility and adding features and capabilities as needed. We remain fully committed to working with partners, including Apple, Facebook, Google, Microsoft and Mozilla to maintain the security and compatibility of Flash content."
I never recommend the SHA1 links with the embedded version numbers because they go dead as soon as a new version is released. Such as http://fpdownload.adobe.com/get/flashplayer/pdc/32.0.0.156/install_flash_player_ax.exe, http://fpdownload.adobe.com/get/flashplayer/pdc/32.0.0.156/install_flash_player.exe, and http://fpdownload.adobe.com/get/flashplayer/pdc/32.0.0.156/install_flash_player_ppapi.exe.