This will be a short post. You might've tried to update your copy of Windows 10 Enterprise, but the updater said you should contact your system administrator in order to update. If you download the full Windows 10 1607 image, it doesn't give you an option to save your files and apps. The only option is the full reinstall.
To fix this issue, I had to temporarily change my serial number to this one:
NPPR9-FWDCX-D2C8J-H872K-2YT43. You can change in your
Settings → Update and Security → Activation app. Never mind that Windows doesn't activate with this serial number — you just need to upgrade.
After I changed my serial number, I just mounted the full 1607 image and ran
Setup.exe from image root. The installer checked for updates and then suggested that I install the update saving all my files and apps, just as you would expect.
After installing the update, change your serial number again to one you legally own.