Legal query regarding defamation case

My cousin was in a relationship with a boy for 8months.She ended the relationship because the boy was cheating on her.One mutual friend asked about their relationship so she told him that they are not together as the boy was cheating on her.This friend revealed this conversation to the boy without informing her.The boy got angry and threatened her that he will put a defamation case on her if she doesn’t apologise over email and even sent a draft format for a formal apology.We did not want to get into any trouble so my cousin wrote the mail saying that her conversation with the mutual friend was not intended to defame him and it was a conversation between two friends.After 2-3 days the boy again wrote her a mail saying that she should keep her mouth shut and he will show the previous mail to all the common people to prove her lie.She is quite afraid as the boy even threatened to come to her house and create a scene.Are there grounds of a case of defamation against my cousin?