Understanding the similar properties of people involved in group search sessions has the potential to significantly improve collaborative search systems; such systems could be enhanced by information retrieval algorithms and user interface modifications that take advantage of important properties, for example by re-ordering search results using information from group members' combined user profiles. Understanding what makes group members similar can also assist with the identification of groups, which can be valuable for connecting users with others with whom they might undertake a collaborative search. In this workshop paper, we describe our current research efforts towards studying the properties of a variety of group types. We discuss properties of groups that may be relevant to designers of collaborative search systems, and propose ways in which understanding such properties could influence the design of interfaces and algorithms for collaborative Web search.