Thanks for the suggestion, it's not built in and while writing this I think ... why not? I suppose this should be added right on top of the @todo list for Groups!
Currently you could achieve it by maintaining your own table (relating users allowed to write and post IDs) and using Groups' API to check if the current user belongs to the group or has the right capabilities needed to edit a post.
Although this doesn't cover the exact requirement you outlined, a group allowed to fully manage a
Pizza custom post type would have these capabilities: