Share a folder from your linux host to your windows guest, e.g. /home/user/GDrive
Install Google Drive in Windows, and configure to sync to your hard drive image, e.g. C:\Users\IEUser\Google Drive, not the shared drive; Google Drive won't sync to a networked drive.
Install SyncToy, and configure it to sync between your Google Drive location and your shared drive, e.g. \\VBOXSVR\GDrive. Use the network name rather than the drive letter assigned to the shared drive for scheduled syncing to work.
Create a scheduled task to run as IEUser whether the user is logged in or not (you don't need to run with the highest privileges). Set the trigger to one time, and repeat every 5 minutes indefinitely. Set the action to run "C:\Program Files\SyncToy 2.1\SyncToyCmd.exe" with the -R argument.