Update Item's Group Using API

I am trying to update the group that an item sits in using the mutate method in the API. I have tried the code below which won’t work as the group isn’t really a column. Does anyone know how I can update the group that an item is in?

mutation {
change_multiple_column_values(item_id:0000, board_id:000, column_values: “{"group" : {"label" : "Done"}}”) {
id
}
}