What about the children that do not have a father to train them or a father that is not willing or able to fulfill that responsibility? 

If we don’t have children’s Bible class how will the (spiritually) fatherless be instructed in the way of the Lord? Given that this is not God’s ideal situation, there is not a quick and easy answer to the problem. However, the scriptures and common sense provide an obvious conclusion… the mother is instrumental in bringing about faith.  

The two gold-standard passages on this would be Timothy, son of a Greek father, learning his faith from his mother and grandmother (1 Tim 1:5). And second is 1 Cor 7:12-14, where the wife of an unbelieving husband, by staying with her husband, maintains holiness for the children. The second passage is a bit mysterious, but I believe the context shows that the mother’s godly life will sanctify the husband and the children.

In all this, I think we see that life lived out in faith in front of our children is the key. Naturally, faithful friends and family members, fathers in faith, and other Christians can and should live among and with these children to demonstrate faith and provide instruction (Phil 2:22, 1 Tim 1:2, Titus 1:4, Titus 2:6-8).