Magento2 – How to Call Default Menu in Static Block

First off, the code should be

{{block type="core/template" template="custom/employee-links-template.phtml"}}

No need to edit your local.xml you can call your static block in your custom/employee-links-template.phtml template, see below.

   <?php if (Mage::getSingleton('customer/session')->isLoggedIn() && Mage::getSingleton('customer/session')->getCustomerGroupId() == "4") :?>
    <?php echo $this->getLayout()->createBlock("cms/block")->setBlockId("your-static-block-id")->toHtml();?>
   <?php endif;?> 

Explanation: You cannot add children to blocks that are dynamically created with {{block}}, or access them from the layout XML, so you have to create the inner block directly from the template.

the best way would be to add some logic to the navigation to handle your additonal links.

However, easiest way would be to just give the links unique classes or ids and then activate them based on the body class:

body.cms-page-view.cms-about .about-us-button {color:red;}

body.cms-index-index.cms-home .introduction-button {color:red;}