Which of the following allows a Windows 7 remote user to edit a file offline then automatically update the changes when back in the office?

Windows Defender is an anti-malware application and has no functionality related to file synchronization or offline editing.

Sync Center manages offline file synchronization between a Windows 7 computer and network shares. When a user edits offline files and returns to the office, Sync Center automatically detects the reconnection and syncs the changes back to the network location, resolving any conflicts as needed.

Windows Aero is a visual interface theme that provides graphical effects such as transparency and animations, unrelated to file synchronization.

HomeGroup is a feature for sharing files and printers among home computers on the same network but does not provide offline file editing or automatic sync.