I'm not sure why you need $name to be a global, but you could try something like this:

use strict;
use warnings;

package abstracted;
our $name = 'parent';

sub get_that_name {
    my $self  = shift;
    my $class = ref $self;
    no strict 'refs';
    return ${"${class}::name"};
}

package abstracted::versionA;
use base 'abstracted';
our $name = 'vA';

package abstracted::versionB;
use base 'abstracted';
our $name = 'vB';

package main;
foreach my $obj (
    map { bless [] => $_ }
        qw/abstracted abstracted::versionA abstracted::versionB/)
{
    print $obj->get_that_name."\n";
}
[download]

output:

parent
vA
vB
[download]

see also symbolic references