synchronizing Contacts between iOS and Android does not work automatically, but can be done manually:
- Export your Google contacts to a vCard database (when in your contact list click on More, Export, in the opening window select all contacts and choose vCard format for export into Apple addressbook)
- Open Contacts (formerly named Addressbook) on your Mac and select Import from the File menu, in the opening FileOpen dialog select the vCard database containing your Google contacts you just created.
- Now synchronize your iPhone with your Mac and the Google contacts will be on your iPhone.
- If you are using Windows instead of OSX the above outlined procedure works different, since you do not export to vCard database, but to an Outlook CSV file, which you then import into Outlook.
Creating and managing groups works with the Contacts app on your Mac or with Outlook on your PC. As far as I know this cannot be done on the iDevice.
Hope this helps,