If we have ROOT access we can manualy backup the file that contains seeds needed to generate password in application.
Backup: You need to set ROOT access for ADB, and then:
adb pull /data/data/com.google.android.apps.authenticator2/databases/databases
The content of the file can be display with help of sqlite client application:
sqlite3 ./databases select * from accounts;
The above method can be done with file manager that can get ROOT access
Restore backup: You need to set ROOT access for ADB, and then:
adb push databases /data/data/com.google.android.apps.authenticator2/databases/databases adb shell #here we need to know which user and group we need to use chown user:group /data/data/com.google.android.apps.authenticator2/databases/databases
The above method can be done with file manager that can get ROOT access and copy databases file to:
/data/data/com.google.android.apps.authenticator2/databases/databases
Also remember to set ownership for proper application user (com.google.android.apps.authenticator2) on this file, because with wrong one (root) the application will crash every time you try to open it.