My mother-in-law is sick and is not expected to recover. Her family is unable to give her expiation for not fasting. My wife, however, has kept some money with me for her mother's shroud and burial costs. Is it permissible to give some of this money as part of the expiation that is obligatory upon her?

All perfect praise be to Allaah, the Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

The sick person whose recovery is not expected is required to give an expiation by feeding a poor person each day. Consequently, your mother-in-law is required to give this expiation. Allaah The Exalted Says (what means): {And upon those who are able [to fast, but with hardship] an expiation [as substitute] of feeding a poor person [each day].} [Quran 2:184]

This is obligatory if she can afford it. However, because Allaah does not charge a soul except with that within its capacity, if she cannot afford paying the expiation, then she will be responsible for nothing as stated by Ibn Qudaamah in Al-Mughni.

If she has a sum of money with you and she does not need it in her necessary expenses, she will be required to give the ransom by feeding a poor person for each day that she does not fast. The amount of that ransom is one Mudd (a volumetric measure [handful], which equals 750 grams) of food.

Saving this sum for her shroud after death does not exempt her from the obligation of giving the expiation because if she dies and leaves no money that can be used in burying her, then the shrouding will be the responsibility of those who have to provide for her, namely, her heirs.

However, you are not permitted to give the expiation on her behalf without her permission. Rather, she should explicitly authorize you to do so.  

Allaah Knows best.