What is the Islamic perspective upon fasting on behalf of someone else? Is it permitted to fast on behalf of somebody else?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Fasting on behalf of someone else who is still alive is invalid. If the person is able to fast, he should do it himself. However, if the person is unable to fast, then his inability is either temporary or lasting. If it is temporary, then this person should wait till he becomes fit again and fast the days he had missed. If his unfitness is lasting, then he is not asked to fast; instead he should feed a needy person for each day (i.e. one meal in the country of the beneficiary) he did not fast as expiation.

As regards fasting on behalf of a dead person, then there are two probabilities:

If the dead person died before he got any opportunity to fast his missed fasting, then nobody in this case is asked to fast on his behalf as the latter was not able to fast when he was alive.
If the dead person had the opportunity to fast but he neglected to do so, then the scholars believe that it is his guardian who should fast on his behalf. For example, if a person was supposed to make up for some days of Ramadan that he missed but he neglected to make them up until the Ramadan of the next year came, and he did not have a sound reason for not making up the fast and he died before making up the missed days, then in this case, it is desirable for his guardian to fast on his behalf. Also, if a person made a vow to Allah to fast and died before fulfilling it, it is desirable for his guardian to fast on his behalf. The evidence for the above two cases is that a man came to the Prophet and said: “My mother had died and she was supposed to fast a whole month, shall I fast on her behalf? The Prophet replied: “Yes”, Allah's debt is worthier to be paid back”. Another Hadith reads: “If anyone dies while he had to make up for some fasts, then his heir must fast on his behalf” [Muslim] So, now it is clear when someone could fast on behalf of someone else.

Allah knows best.