You can create an auxiliary executable (be very careful, it is dangerous) which any user can run it with root permissions.
This is called setuid.
For safety reasons, you cannot setuid
a shell script.
Extracting from the wiki how to use it:
The setuid and setgid bits are normally set with the command chmod by setting the high-order octal digit to 4 (for setuid) or 2 (for setgid). "chmod 6711 file" will set both the setuid and setgid bits (2+4=6)
Update
As @rici noted, you still will need to have execution permission to execute this process, so you can remove execution permission from others
and keep it only on group
. So, only who is member of the group can execute it.