One of the members of the group can generate a new key pair, and share it with all the other members (in the same way one would share a file with each individual). If Alice sends a file to the group's key pair, anyone in the group can read the file.

Adding members to the group is trivial; just send them the group's key pair. Removing a member would be more difficult. Perhaps the most convenient way would be to add an additional layer of security on top (so members would need server access permissions, plus the private key). The only other option would be to create a new group and to re-encrypt all the existing files with a new key.