Is it considered a sin to give the Zakat Al-Fitr received by the mother—who earns a large salary—to the sister whose husband has passed away and is needy, having no source of income except a pension of 80 pounds per month, without the mother's knowledge?

If a man paid Zakat (definition) al-Fitr to your mother, and she is not eligible, and you took this Zakat without your mother's knowledge and gave it to your eligible sister, then scholars differ regarding someone who pays Zakat to a wealthy person, assuming they are poor. Some believe it is valid, while others do not.

If we say it is valid, then the man's obligation is discharged, and the money becomes your mother's property. Your disposal of it without her permission necessitates you guaranteeing it to her, unless she absolves you.

If we say it is not valid, then if the man did not clarify that it was Zakat al-Fitr, he is not entitled to reclaim the money from your mother, and you must guarantee it to your mother. However, if he did clarify that it was Zakat al-Fitr, then the money must be returned to him so he can pay it to its rightful recipient, or you return it to your mother so she can return it to him.

In both cases, your handling of the money was incorrect, and you are liable for it. As for what you paid to your sister, you can reclaim the money from her or leave it for her as charity.