Variable inside class (whose value is derived from another class) isn't available to methods

Variable inside class (whose value is derived from another class) isn't
available to methods

I'm new to OOP so there's probably something fundamental I'm missing here.
In the Foo class, the $this->user_group variable is not available to the
bar() method. Why is that? And is there a less messy way to get these
classes to all talk to each other than including the $DB (and others)
every instantiation (this gets hairier with my actual code, which has many
more classes). Here's an example:
class Foo {
private $Auth, $user_group;
function __construct($DB) {
$this->Auth = new Auth($DB);
$this->user_group = $this->Auth->get_user_permissions_group();
// ("echo $this->user_group;" here will output the correct value)
}
public function bar() {
// ("echo $this->user_group;" here will output nothing)
return ($this->user_group > 1 ? 'cool!' : 'not cool!');
}
}
class Auth {
private $DB;
function __construct($DB) {
$this->DB = $DB;
}
public function get_user_permissions_group() {
$result = $this->DB->query('return user permissions level from DB');
return $result; // int, 1-3
}
}
$DB = new Database();
$Foo = new Foo($DB);
echo $Foo->bar();