diff --git a/core/lib/Drupal/Core/DrupalKernel.php b/core/lib/Drupal/Core/DrupalKernel.php
index 7ab72da..22c7097 100644
--- a/core/lib/Drupal/Core/DrupalKernel.php
+++ b/core/lib/Drupal/Core/DrupalKernel.php
@@ -897,7 +897,9 @@ protected function initializeCookieGlobals(Request $request) {
       // Build an unique session name based on host and any site directory. We
       // avoid including the protocol to use the same session identifiers across
       // HTTP and HTTPS.
-      $session_name = $request->getHost() . $request->getBaseUrl();
+      $session_name = $request->getHost() . $request->getBasePath();
+      // Replace core out of session name so core
+      $session_name = preg_replace('/\/core$/', '', $session_name);
 
       // HTTP_HOST can be modified by a visitor, but has been sanitized already
       // in DrupalKernel::bootEnvironment().