What if the mother lied to protect her child and herself? whatever the reasons the mothers have,, it her business. What if the father have a criminal history? What if he done something to her? maybe she have a good reasons to lie. Some women I have met says they did it for protection of their children. But, hey I respect their deiscions and their business.
There are laws to help protect children, have them call the police if a child/mother are in danger and get a lawyer.
Otherwise they will get caught in one of their lies.