Can a woman who is breastfeeding fast or not?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

A suckling mother is permitted to break fast if she fears for herself or her child because of fasting. Then, she has to make up for the missed days. The Prophet (Sallallahu Alaihi wa Sallam) said: "Allah, The Great and Almighty, has relieved the traveler of fasting and half of the prayer (i.e. to shorten it), and the pregnant and the breast-feeding woman of the fast" [ Ahmad ]. But, if she does not fear, then she is not allowed to break fast.
As for feeding the child with artificial milk so that a mother can fast, there is no harm as long as this does not constitute any harm to the child. Here, a mother is required to fast, since there is no legal reason to break fast.
Allah knows best.