Unfortunately it is a lesson that the child must learn for him herself.  Making false accusations such as that must bear some weight, or children who really are being abused fall by the way sside.  It is unfortunate, but eventually the child making such accusations will learn that the only person being punished is him/herself by being separated from the family.  The Mother must remain strong and determined.  Some children MAY be more emotional, but they need to learn just the same that there is a consequence to everything.  The minute the mother shows the child that she is bothered by this whole performance, the child will technically have a means by which she can control the mother.  It is best to just leave things for a while in my opinion until the child calms down and realises that no one is rushing to recover her from the strangers home.  you can bet ten to one that a bit of reverse psychology will sort this matter out and it may not occur a thrid time.  TOUGH LOVE.  The Mother should however make it quite clear to authorities that this is what she is doing and that her lack of action is nothing to do with being or feeling guilty. (This is exactly what I would do myself anyway in such instance).