How to move contacts from one group to another?
Add the same accounts to you Mac and make sure you have the "Contact" selection checked for each account.

This will sync all your contacts in the Contacts app for each email account. Open the Contacts app and you will be able to see each individual email account (on the left side) and its associated contacts. 

Click on the account you wish to transfer from. Drag-and-drop desired contacts from one email account to the other. This effectively transfers your contacts to the account you want. Deleting and adding to old and new email accounts respectively.

As far as I know, there is no way to do this in iOS.