but I won't stop with that

Here is one way of doing it
input hash

my %hash= (
        Group1 => 'ATG1,ATG2,ATG4,ATRG7',
        Group2 => 'ATG1,ATG9',
        Group3 => 'FYCO1,LSM2',
        Group4 => 'ATG1,MAM2',
        Group5 => 'LSM2',
        );
[download]

use strict;
use warnings;
use Data::Dumper;

my %hash= (
        Group1 => 'ATG1,ATG2,ATG4,ATRG7',
        Group2 => 'ATG1,ATG9',
        Group3 => 'FYCO1,LSM2',
        Group4 => 'ATG1,MAM2',
        Group5 => 'LSM2',
        );
my (%newhash,$i,%combinegroups,%splittedhash);
foreach my $key (keys %hash){
    my @values ;
    if($hash{$key} =~ /,/){
        @values = split(/,/,$hash{$key}) ;
        $splittedhash{$key} = [ @values ];
    }else{
        push(@values,$hash{$key});
        $splittedhash{$key} = [ @values ];
    }
    if(++$i == 1) {
        my @newvalues;
        push (@newvalues,$key)  for(0..scalar @values);
        @newhash{@values} = @newvalues;
    }else{
        foreach (@values){
            if(exists $newhash{$_}){
                $combinegroups{$newhash{$_}.":".$key}++;
            }else{
                $newhash{$_} = $key;
            }
        }
    }
}
foreach my $key (keys %combinegroups){
    my ($firstgroup,$secondgroup) = split(/:/,$key);
    my %unique;
    @unique{@{$splittedhash{$firstgroup}},@{$splittedhash{$secondgroup
+}}} = ();
    $hash{$firstgroup} = join(',',keys %unique);
    @{$splittedhash{$firstgroup}} = keys %unique;
    delete $hash{$secondgroup};
}
print "FINAL HASH\n",Dumper \%hash;
[download]

FINAL HASH
$VAR1 = {
          'Group4' => 'ATG9,ATG2,ATRG7,ATG4,ATG1,MAM2',
          'Group5' => 'FYCO1,LSM2'
        };
[download]

I haven't worried about efficiency in my code, it can be written even more effectively!, but this will work for your this case
if a value from 1 group (key) is found as a value from another group then I want to combine the values from both of those groups into a single set of values