When the form is processed, all checked boxes will be
       returned as a list under the parameter name 'group_name'.
       The values of the "on" checkboxes can be retrieved with:

             @turned_on = $query->param('group_name');
[download]

This does not seem to be the case, the following code

#!/usr/bin/perl -w
use strict;
use CGI;

my $cgi = new CGI;
print $cgi->header();
print $cgi->start_html();
my $submit = $cgi->param('submit');
if ($submit)
{
    my @on = $cgi->param('desig');
    foreach my $item (@on)
    {
        print qq($item<BR>);
    }
    exit;
}
print $cgi->startform(-method=>'post');
print $cgi->checkbox_group(-name=>'desig',
                -values=>['a', 'b', 'c', 'd', 'e', 'f', 'g']);
print $cgi->submit(-name=>'submit', -value=>'Submit');
print $cgi->endform();
print $cgi->end_html();
[download]

yields

a
c
e
g
[download]

when every other box is checked

Only if you're using a broken/proprietary CGI parameter parsing routine that does something like this:

foreach my $pair (split /&/, $ENV{QUERY_STRING}) {
   my ($key, $value) = split(/=/, $pair);
   $parameters{$key} = $value;
}
[download]

The correct solution is to defer processing like this to CGI.pm, which will correctly handle multiple checkboxes with the same name.