It IS possible.
Check out this code (valid C89, heh) I wrote in 25 mins. Usage example:
whitequark@forth:~/skipgroup$ ls
skipgroup.c
whitequark@forth:~/skipgroup$ gcc skipgroup.c -o skipgroup
whitequark@forth:~/skipgroup$ sudo chown root:root skipgroup
whitequark@forth:~/skipgroup$ sudo chmod u+s skipgroup
whitequark@forth:~/skipgroup$ groups
whitequark adm dialout cdrom plugdev lpadmin admin sambashare
whitequark@forth:~/skipgroup$ ./skipgroup
Usage: ./skipgroup <group to remove>
Must be SUID. Launches shell.
whitequark@forth:~/skipgroup$ ./skipgroup cdrom
$ id
uid=1000(whitequark) gid=1000(whitequark)
groups=4(adm),20(dialout),46(plugdev),104(lpadmin),114(admin),118(sambashare),1000(whitequark)
WARNING THIS CODE IS SUID!
While it drops privileges as you see on id's output, it MAY BE DANGEROUS. Dixi.