Why do I get my own text message sent back to me in a group message on iphone 5S?
Several things can cause this.  I would need more information to give you an accurate answer. If you are using iMessage it could be that a number or email in the group is somehow associated with your contact information.  Go to Settings>Msessages and make sure that only one phone number and AppleID is linked to your account. Check the group and make sure you only see your email or your phone number in the list, not both.

If it's SMS contact your carrier as that is controlled by the carrier programing.