Is Zakat al-Fitr obligatory for a child the father has not seen since his apostate mother fled with him?

The father must pay Zakat al-Fitr on behalf of his absent son, because the original assumption is that he is still alive and that he is Muslim, following the Islam of his parents. His Fitr is obligatory upon the father until the son reaches an age where he can earn a living. The father is also obligated to search for his son and raise him upon Islam and teach him his duties.