Most imams in America do not know that khula is given by an imam or Islamic authority on the appeal of the wife. It is not given or controlled by the husbands.
Why are these imams mistaken? One reason is that they don’t know the correct audience of the verse of the Qur’an about khula. Here it is:
“Then if you fear that they would not be able to keep the limits ordained by Allah, then there is no sin on either of them if she gives back (the Mahr or a part of it) for her Al-Khul‘ (divorce)”
Notice the word “you” [singular] referring to the religious authority and “they” [plural] referring to the husband and wife.
Many imams make the blunder of thinking that “you” is the husband, which, if true would make no sense of “they.”
Is there any evidence in the Qur’an or the hadith that the husband can give khula?
Absolutely not, He can give talaq [for which there are numerous rules], but not khula.
Is there evidence that only the imam or religious authority can give khula?
One example is the hadith of the Prophet, pbuh, according to which he gave khula even when the woman did not have any evidence of brutality or abuse by the husband. 
It was only on the basis of incompatibility.
Another example is that of Umar, r.a., who gave khula to a woman about whom he could not understand why she wanted khula.’ He shut her in a dark room and when she was released, she said that was the first time she had slept in peace. 
It was only on the basis of incompatibility. These were good husbands and good Muslims who would never mistreat their wives.
The third Righteous Caliph, did not want anything but a symbolic gesture of the return of gifts by the wife in khula. Just a hair pin.
There is also evidence that the wife can initiate khula without going to an imam. If the husband does not accept it, then she can go to an authority. 
The husband under Saudi rulings can control khula by asking for writing down of khula and signatures by two witnesses.
Again, there is NO evidence that writing or witnesses are required in khula. Neither the Prophet, pbuh, nor Umar, r.a., nor Usman, r.a., carried out writing or witnessing.
1. Volume 7, Book 63, Number 199: Narrated Ibn ‘Abbas:
The wife of Thabit bin Qais bin Shammas came to the Prophet and said, “O Allah’s Apostle! I do not blame Thabit for any defects in his character or his religion, but I am afraid that I (being a Muslim) may become unthankful for Allah’s Blessings.” On that, Allah’s Apostle said (to her), ‘Will you return his garden to him?” She said, “Yes.” So she returned his garden to him and the Prophet ordered him to divorce her.
2. Ibn Jarir, narrating from Kathir, maula of Samra,
3. Abdur Razzaq narrating about Usman, r.a. about Rabia.
4. Also from Usman, the story of Rabia daughter of Mauz bin ‘Afra.
Independent scholars are quite clear about Khula. For example:
“What is khula? When a woman does not like her husband for whatever reason and gets rid of him by returning his marriage gift to him.” [Minhaj al-Muslim by Abu Bakr Jabir al-Jazairi, fadeelat al-Shaykh, instructor at the Mosque of the Prophet, pbuh, in Madinah