Praise be to Allah the Lord of the Worlds.
It is forbidden for the rich person to ask others to give him/her Sadaqa (Voluntary charity) and whatever he/she takes through that means is unlawful. In fact, Islam has warned against this in several Hadiths, some of which are the following: Ibn ’Umar (RAA) narrated that The Messenger of Allah (PBUH) said: “A man who persists in begging people to give him charity, will come on the Day of Judgment and there will not be a piece of flesh on his face.” Agreed upon."{Bukhari & Muslim}. Also, it was narrated that Abu Hurairah said: "The Messenger of Allah said: 'It is not permissible to give charity to a rich man (or one who is independent of means) or to one who is strong and healthy."'{Muslim}.
Moreover, Abu Hurairah (May Allah be pleased with him) reported: Messenger of Allah (PBUH) said, "He who begs to increase his riches is in fact asking only for a live coal. It is up to him to decrease it or increase it." {Muslim}.
As for the giver, if he gave the Sadaqa under the assumption that the recipient is in need, then Allah will reward him/her for it. The evidence is the following Hadith: Ma'n bin Yazid bin Akhnas (May Allah be pleased with them) (he, his father and his grandfather, all were Companions) reported: My father set aside some dinars for charity and gave them to a man in the mosque. I went to that man and took back those dinars. He said: "I had not intended you to be given." So we went to Messenger of Allah (PBUH), and put forth the matter before him. He said to my father, "Yazid, you have been rewarded for what you intended." And he said to me, "Ma'n, you are entitled to what you have taken." [Al- Bukhari].If the given sum was Zakat {Obligatory charity}, then the giver is allowed to take it back to give it to one of the eligible recipients prescribed in the Noble Quran to be cleared from the liability before Almighty Allah. although it was Sadaqa, then he isn`t allowed to take it back, even if the taker didn`t seem in need, because it is allowed to give Sadaqa to the rich, and Allah will reward him for that.
Ibn Mofleh (May Allah have mercy on him) said, "Ibn Shihaab said, "The giver of Sadaqa isn`t obliged to take it back if he realized that the recipient is rich.
This is because he/she gave it for the reward and since he/she has attained that, he/she isn`t allowed to take it back. On the other hand, if it was Zakat, then he/she is allowed to take it back to give it to one of the eligible recipients specified in the Noble Quran to clear himself/herself from the liability before Almighty Allah."{Al-Furu` Wa Tas-hih Al-Furu`, 4/294}.
In conclusion, the money that the asker has taken back is unlawful and she shouldn`t have resorted to deception to do that. But, since she has taken it back, she doesn`t have to return it to that beggar. And Allah the Almighty knows best.