Final answer:
The effective group ID (GID) of a process is set using the SGID bit in the file permissions. Option '4744' correctly sets this bit, allowing executed files to run with the group permissions of the file's owner.
Step-by-step explanation:
The effective group ID (GID) of a process can be set using the permissions on a file when the set-group-ID (SGID) bit is set. The permissions in the Unix and Linux file systems are denoted by a three or four-digit octal number. Of the options provided, 2666 does not set the SGID bit, and 777 sets full permissions without setting the SGID bit. However, 4744 is the permission set where the first digit '4' activates the SGID bit along with setting the file owner's permission to read (4), write (4), and execute (4).
An SGID bit set on a file will cause a process that runs that file to have its effective group ID set to the group ID of the file. To clarify, when a script or program with SGID is executed, it runs with the group permissions of the group that owns the file, not the group of the user who ran it. This is particularly useful for allowing users to execute programs with the permissions of a specific group they may not necessarily belong to.