...which is not working as expected.

So what do you expect?  Basically, the approach is fine — except that you got it the wrong way round wrt what is child and parent (if ($pid == 0) {...} is the child).

Here's a slightly modified version which does what I would expect, i.e. have perl.pl (and its children, if any) run under a new process group:

#!/usr/bin/perl

use strict;
use warnings;

my $cmd = qq{nohup ./perl.pl &};

my $pid = fork;  die 'fork failed' unless defined $pid;

if ($pid) {
    print STDERR "parent pid=$$ child pid=$pid\n";
    wait;
    my $sid = getppid;
    system "ps f -s $sid -o pid,ppid,pgrp,sid,cmd";
} else {
    print STDERR "child pid=$$ \$pid=$pid\n";
    setpgrp;
    exec $cmd or die "Bad exec: $!";
}
[download]

With perl.pl being

#!/usr/bin/perl

system 'sleep 1';
[download]

the output is something like

$ ./813283.pl
child pid=31851 $pid=0
parent pid=31850 child pid=31851
  PID  PPID  PGRP   SID CMD
18307 18305 18307 18307 bash -rcfile .bashrc
31850 18307 31850 18307  \_ /usr/bin/perl ./813283.pl
31853 31850 31850 18307      \_ ps f -s 18307 -o pid,ppid,pgrp,sid,cmd
31852     1 31851 18307 /usr/bin/perl ./perl.pl
31854 31852 31851 18307  \_ sleep 1
[download]

As you can see in the PGRP column, perl.pl and its child process sleep 1 is running under a different process group (31851 in the sample output).