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Since we have to support Joomla versions from 3.0 to 3.5 we can't have an HTTP code selection box as this feature would be inconsistent. In the future we may add such a feature or change the HTTP code.

Actually, looking at the code again, your solution is WRONG. The problem comes from the two different generations of Joomla core code (3.0 to 3.3 and 3.4 to 3.6).

In the first generation where JApplicationSite extends JApplication the method's signature is

public function redirect($url, $msg = '', $msgType = 'message', $moved = false)

In the second generation where JApplicationSite extends JApplicationCms the method's signature is

public function redirect($url, $status = 303)

However, as you can read in libraries/cms/application/cms.php lines 974 to 1014 you can still use the four arguments call. In this case Joomla! will translate your arguments to the new style and then call parent::redirect(). In its turn, parent::redirect() is actually a call to JApplicationWeb::redirect(). However, as we can see in JApplicationCms line 1012 Joomla has a bug and does NOT translate the $moved boolean to an HTTP code integer. So, basically, you are reporting a Joomla! bug.

So despite us using the Joomla API per its documentation (docblocks), a bug in Joomla itself prevents the code from working properly. Since we have to support older versions of Joomla which do not support the new, two argument call to redirect() you'll have to live with it until we drop support for old Joomla versions – in the next version of Admin Tools, scheduled for late June to mid July.

It shouldn't work in Joomla! 3.5.1. Not unless you change that line to $this->app->redirect($targetURL, 303);

So, basically, the TESTED change you have to make is to replace that line with this:

			if (class_exists('JApplicationCms') && class_exists('JApplicationWeb')
				&& ($this->app instanceof JApplicationCms)
				&& ($this->app instanceof JApplicationWeb))
			{
				$this->app->redirect($targetURL, 303);

				return;
			}

			$this->app->redirect($targetURL, '', 'message', true);