Hallo Frank!
Thank you for your 2 statements.
This is my test plattform:
V-Server from Strato.AG (in Berlin) / My Zen-Shop in Stuttgart / There is no time difference in between. Die Zeitzone zum Ausführen der Aufgabe ist (UTC +01:00) Europe / Berlin
Server Betriebssystem: Linux 3.13.0-042stab111.12 Datenbank: MySQL 5.5.47-0ubuntu0.14.04.1
Server Datum: 15.02.2016 13:09:07 Datenbank Datum: 15.02.2016 13:09:07
Server Up Time: 13:09:07 up 2:27, 0 users, load average: 0.14, 0.06, 0.03 HTTP Server: Apache
PHP Version: 5.5.9-1ubuntu4.14 (Zend: 2.5.0) PHP Memory Limit: 256M PHP Safe Mode: Off
PHP Datei Uploads: On Max. Größe: 16M POST Max. Größe: 16M

My shop is to see under: lieferservice.restaurant-weilerhau.de
Now I did (once more than 100 times) the same as in installation documentation and after your statement:
1.Install „auto_store_open_close.php“ (today: auto_store_open_close.php, v 1.2 2016/02/15 / code edited direct in cPanel , no copy per FTP)
2.Than, I’ve uploaded the „init_cron_admin_user_id.php“ file in /includes/init_includes/overrides/
3.Cron jobs:
If I write the cron job with „wget“ like:

30 11 * * * wget http://My_Domain/zencart/zen_admin/a...open_close.php
00 22 * * * wget http://My_Domain/zencart/my_zen_admi...open_close.php

I do not get any answer……

If I comand:
lynx -source http://www.restaurant-weilerhau.de/z...open_close.php -auth=?Skalkos$2015
every day at 11.30

The Apache server starts correctly his cron job and send
allways same report per Email:

regarding: Cron <skalkos@h2184067> lynx -source http://www.restaurant-weilerhau.de/z...open_close.php -auth=?Skalkos$2015

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" dir="ltr" lang="de">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Zen-Cart 1.5</title>
<link href="includes/stylesheet.css" rel="stylesheet" type="text/css" />
<meta name="robots" content="noindex, nofollow" />
<script language="javascript" type="text/javascript"><!--
function animate(f)
{
var button = document.getElementById("btn_submit");
var img = document.getElementById("actionImg");
button.style.cursor="wait";
button.disabled = true;
button.className = 'hiddenField';
img.className = '';
return true;
}
//--></script>
</head>
<body id="login" onload="document.getElementById('admin_name').focus()">
<form id="loginForm" name="loginForm" action="http://restaurant-weilerhau.de/zencart/zen_admin/login.php?camefrom=auto_store_open_close.php" method="post" onsubmit="animate(this)">
<fieldset>
<legend>Admin Login</legend>
<label class="loginLabel" for="admin_name">Benutzername: </label>
<input style="float: left" type="text" id="admin_name" name="admin_name" value="" autocomplete="off" />
<br class="clearBoth" />
<label class="loginLabel" for="admin_pass">Passwort: </label>
<input style="float: left" type="password" id="admin_pass" name="admin_pass" value="" autocomplete="off" />
<br class="clearBoth" />
<p class="messageStackError"></p>
<input type="hidden" name="securityToken" value="f2563929bffddfc95bdb352566d9cff0">
<input type="submit" name="submit" class="button" value="Login" id="btn_submit"/>
<input type="hidden" name="action" value="dof2563929bffddfc95bdb352566d9cff0" id="action1"/>
<img id="actionImg" src = "images/loading.gif" class="hiddenField" />
<br /><a style="float: right;" href="http://restaurant-weilerhau.de/zencart/zen_admin/password_forgotten.php">Passwort vergessen?</a>
</fieldset>
</form>
<div id="loginExpiryPolicy">LOGIN_EXPIRY_NOTICE</div>
</body>
</html>


By the way:
1. wget is one of the files that needs to have rights 700, else every hackers uploaded script can use it to download the rest of the hack from external websites.
2. “lynx” - advantage is that the php file is executed with the rights of the website user and not root.
the "-auth=loginP asswd" is the login info because the cron.php is placed inside a password protected directory.
An extra advantage of running a cron with lynx is that it is executed with not only the rights, but also the configuration of that website.
If the site has a custom php.ini then it will be run with those settings.
Do you think that this could be a reson, why the “„init_cron_admin_user_id.php“ in our case is not doing well?

I’m desolate, unfortunately I can not get it work.
let me know if you see any mistake.
Thank you and have a nice night.
Best regards, Mihaela