Upon attempting to update a fresh Windows 7 SP1 installation I encountered error 80072EFE:
Most information regarding this error code available on the web at present writing indicates some type of connection error. Despite a working connection and ability to browse the web, running the Windows Update Troubleshooter complained that Windows was unable to automatically detect proxy settings for the current network. There being none, and no change in the situation after connecting to a separate network using a different interface, I went looking for updates to the default post-installation set of root certificates and then updates that might enable support, if it was somehow missing from the update service's capabilities, for the new .ms TLD. Stopping the Windows Update Service and deleting its associated cache and temporary files also proved fruitless.
The solution in my case was to manually update the Windows Update Agent per https://docs.microsoft.com/en-us/troubleshoot/windows-client/deployment/update-windows-update-agent.
The (current at time of writing) download link for Windows 7 x64 is http://download.windowsupdate.com/windowsupdate/redist/standalone/7.6.7600.320/windowsupdateagent-7.6-x64.exe.
I subsequently added this to my trusty customized Windows 7 installer USB stick in case it is encountered again and not related to the particular platform and initial driver pack I was working with (fingers crossed).