Just to say we have taken my friends daughter abroad with us twice (although only to European destinations) and both times have taken a letter from her mum clarifying that she has permission to travel. This could be used in the event of being asked by customs that you have permission to take the child abroad. It also states that should any medical decisions need to be made that her mum is quite happy for us to authorise that too.
Chances are you won't need the letter anyway but better to be safe than sorry. We were on the Eurostar coming back from France a couple of years ago and a family had a child with them who wasn't related, the customs tore a strip off them for not having a consent letter from the childs parents. Not something we'd given much thought to before that!