The novel is even more shocking and revealing.
In it, a Christian Fundamentalist group is responsible. While you might at first think about Jehovah's Witnesses, because of the political intrigue involved, that quickly gets tossed out the window. The ideals of American Protestant Conservatism which still cling to their Puritan roots is actually what is being portrayed. It is what would happen if a group like America's Moral Majority militarized and organized a coup. Puritanism would be reinstated since American Protestant Conservatism is still based on its theology. The theocracy of this story is also governed by the political-socio rules of a military dictatorship.
In the novel groups like Catholics and Jews are executed--Catholics outright, Jews given a "fake" choice to convert and join Gilead (which is what they now call the country) or leave for Israel, during which they are killed while on the journey.
Though no mention is made of NRMs like Jehovah's Witnesses, Seventh-Day Adventists and Mormons, the impression given is that they either assimilate or are quickly executed if they didn't get a chance to escape. Groups that do not support the military and do not submit to the Puritan theology (which the NRMs do not) are dissolved in this dystopian society.