We study the group renaming problem, which is a natural generalization of the renaming problem. An instance of this problem consists of n processors, partitioned into m groups, each of at most g processors. Each processor knows the name of its group, which is in {1, . . . , M}. The task of each processor is to choose a new name for its group such that processors from different groups choose different new names from {1, . . . , }, where < M. We consider two variants of the problem: a tight variant, in which processors of the same group must choose the same new group name, and a loose variant, in which processors from the same group may choose several different names. Our findings can be briefly summarized as follows: