I'm trying my best to name real-life organizations that match my real-life experience here, and you're seemingly trying to come up with hypothetical examples that may or may not have any relevance to the real world.

Could you give me an example of these organizations that are giving you such trouble in classifying hate group vs not-hate group?

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Like, who exactly are you trying to describe? Are you calling Militant Atheists a hate group (or a candidate for hate groups?) The Satanic Temple?

I'm certainly aware of militant atheist groups if that's where you're trying to go. All of them are "anti-religion", but I wouldn't call any of them a hate group. As a Catholic myself, I've had to deal with their anti-religion and anti-Christian bias. But I've never felt personally threatened, or physically threatened by them.