How do I back up Google Authenticator?
The following method will only work rooted Android devices.

Linux:

adb pull /data/data/com.google.android.apps.authenticator2/databases/databases /AFolderOnPC

Windows:

adb pull /data/data/com.google.android.apps.authenticator2/databases/databases C:\AFolderOnPC

Note that the folder on the PC has to already exist.

This will copy the authenticator database files with the main keys, from which the One Time Passwords are generated, to the PC. The file can then be restored to the same location, on Android devices, or read with an sqlite database viewer to extract the keys.