She insisted on asking him for a divorce so he divorced her without really intending it
If the husband clearly uuters the word of divorce (talaaq), then it counts as a talaaq, whether he intended it or not, because in the case of a clear utterance of divorce it is not stipulated that there should be the intention, unless he was forced to do it or he said it in a state of extreme anger, in which case his divorce does not count as such.
The wife’s persistence in asking for a divorce does not count as being forced. If the husband wanted to put a stop to her persistence and uttered the word of divorce without intending it as such, it still counts as a divorce.
If a man has divorced his wife three or four or five times, with words such as “you are divorced (anti taaliq) three times” or “five times,” or he repeated the word and said “You are divorced, divorced, divorced (anti taaliq, taaliq, taaliq)”, it counts as one divorce according to the more correct view. See the answer to question no. 96194
And Allah knows best.