Rose i am sorry to but in but i think i can help clear up the confusion and One correct me if i am wrong but this is how i see it. Betas have a lot of responsibility's temp leader and adviser to the Alpha is one like you said but this responsibly would be more for the Lieutenant. However just like they may be times when the Alpha cant carry out his responsibility's their may be time when the Lieutenant cant ever so then the other Betas would pick up the slack or when the Lieutenant is acting as the temp leader when the alpha is away then the other Betas would advise him. This is not the only role for a Beta as they oversee Sentinels Hunters and Scouts Ect who all report to the Betas. They also lead hunts with the hunters and get involved with the security issues with the Sentinels and direct the scouts on what to do when they spot something major. These are all things the Lieutenant would not have time for so i think it will work like the Alpha oversees the Lieutenant the Lieutenant oversees all the Betas and the Betas oversees everyone else. 

Hope this helps