Here's a successive character-by-character approach that might work for you:

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

my %groups = (
    group1 => ['cooling', 'rooting', 'hooting', 'looking', 'doormat', 
+'cooking', 'cookies', 'noodles'], # desired result "-oo----"
    group2 => ['yourself', 'southern'], # desired result "-ou--e--"
    group3 => ['arden', 'arlen', 'asked'] # desired result "a--e-"
);

my %matches;
for my $group (keys %groups){
    my $last;
    for my $word (@{$groups{$group}}){
        if (!defined $last){
            $last = $word;
            next;
        }
        else {
            die "Length mismatch ('$last'/'$word')\n"
                unless length($last) == length($word);
            my @chars = split //, $word;
            for (0..$#chars){
                substr $last, $_, 1, "-"
                    unless $chars[$_] eq substr $last, $_, 1;
            }
        }
    }
    $matches{$group} = $last;
}

print "$_: $matches{$_}\n" for sort keys %matches;
[download]

Output:

group1: -oo----
group2: -ou--e--
group3: a--e-
[download]