The Southern Poverty Law Center's definition of a "hate group" includes those having beliefs or practices that attack or malign an entire class of people, typically for their immutable characteristics.
According to the United States Federal Bureau of Investigation, a hate group is any group who's "primary purpose is to promote animosity, hostility, and malice against persons belonging to a race, religion, disability, sexual orientation, or ethnicity/national origin which differs from that of the members of the organization
By definition the Facebook Group "Fuck Israel" is a hate group. Take Action!