Can karma operate at the level of groups i.e. collectively?
Yes, of course, there can be group karma.

To begin with, individual karma is only a useful approximation. Because of anatta, what we conventionally call individual is merely an assembly point of contributing factors. It is these factors that are referred to as karma. They don't belong to anyone, really.

As per the Hierarchy Theory, useful notions of groups can be arbitrarily designated at any level of observation. Individual being, family, social group, country etc. -- all of these are abstractions. Some of the karmic patterns make better sense when looked from far away, in which case it may be helpful to attribute them to a group.