Use a label on your while loop:

HASH: while(my ($key, $value) = each %ids) {
    for my $item (@names) {
        if($item =~ m/$key/) {
            push @groups, $item;
            next HASH;
        }
    }
}
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Alternatively, you could also use last to exit the inner loop and let things take their natural course, as it were.