diff --git a/pyiron_base/jobs/flex/pythonfunctioncontainer.py b/pyiron_base/jobs/flex/pythonfunctioncontainer.py
index 43cb5c33b..01ad1c6aa 100644
--- a/pyiron_base/jobs/flex/pythonfunctioncontainer.py
+++ b/pyiron_base/jobs/flex/pythonfunctioncontainer.py
@@ -44,6 +44,7 @@ def __init__(self, project, job_name):
         super().__init__(project, job_name)
         self._function = None
         self._executor_type = None
+        self._automatically_rename_on_save_using_input = True
 
     @property
     def python_function(self):
@@ -64,23 +65,32 @@ def __call__(self, *args, **kwargs):
     def to_hdf(self, hdf=None, group_name=None):
         super().to_hdf(hdf=hdf, group_name=group_name)
         self.project_hdf5["function"] = np.void(cloudpickle.dumps(self._function))
+        self.project_hdf5["_automatically_rename_on_save_using_input"] = (
+            self._automatically_rename_on_save_using_input
+        )
 
     def from_hdf(self, hdf=None, group_name=None):
         super().from_hdf(hdf=hdf, group_name=group_name)
         self._function = cloudpickle.loads(self.project_hdf5["function"])
+        self._automatically_rename_on_save_using_input = bool(
+            self.project_hdf5["_automatically_rename_on_save_using_input"]
+        )
 
     def save(self):
-        job_name = self._function.__name__ + get_hash(
-            binary=cloudpickle.dumps(
-                {"fn": self._function, "kwargs": self.input.to_builtin()}
+        if self._automatically_rename_on_save_using_input:
+            job_name = self._function.__name__ + get_hash(
+                binary=cloudpickle.dumps(
+                    {"fn": self._function, "kwargs": self.input.to_builtin()}
+                )
             )
-        )
-        self.job_name = job_name
-        if job_name in self.project.list_nodes():
-            self.from_hdf()
-            self.status.finished = True
-        else:
-            super().save()
+
+            self.job_name = job_name
+
+            if self.job_name in self.project.list_nodes():
+                self.from_hdf()
+                self.status.finished = True
+                return  # Without saving
+        super().save()
 
     def run_static(self):
         if (
diff --git a/tests/flex/test_pythonfunctioncontainer.py b/tests/flex/test_pythonfunctioncontainer.py
index b08662c72..595498a8c 100644
--- a/tests/flex/test_pythonfunctioncontainer.py
+++ b/tests/flex/test_pythonfunctioncontainer.py
@@ -55,7 +55,9 @@ def test_with_executor(self):
             self.assertIsNone(job.server.future.result())
             self.assertTrue(job.server.future.done())
 
-    @unittest.skipIf(os.name == "nt", "Starting subprocesses on windows take a long time.")
+    @unittest.skipIf(
+        os.name == "nt", "Starting subprocesses on windows take a long time."
+    )
     def test_terminate_job(self):
         job = self.project.wrap_python_function(my_sleep_funct)
         job.input["a"] = 5
@@ -100,3 +102,49 @@ def test_with_internal_executor(self):
         job.run()
         self.assertEqual(job.output["result"], [6, 8, 10, 12])
         self.assertTrue(job.status.finished)
+
+    def test_name_mangling(self):
+        def make_a_simple_job():
+            job = self.project.wrap_python_function(my_function)
+            job.input["a"] = 1
+            job.input["b"] = 2
+            return job
+
+        job = make_a_simple_job()
+        self.assertEqual(
+            job.job_name,
+            my_function.__name__,
+            msg="Sanity check"
+        )
+        try:
+            job.save()
+            self.assertNotEqual(
+                job.job_name,
+                my_function.__name__,
+                msg="By default, we expect the wrapped job names to get mangled based "
+                    "on their input so multiple calls to the wrap get unique names"
+            )
+            loaded = self.project.load(job.job_name)
+            self.assertTrue(
+                loaded._automatically_rename_on_save_using_input,
+                msg="The mangling preference should survive saving and loading"
+            )
+        finally:
+            job.remove()
+
+        job = make_a_simple_job()
+        job._automatically_rename_on_save_using_input = False
+        try:
+            job.save()
+            self.assertEqual(
+                job.job_name,
+                my_function.__name__,
+                msg="When requested, the job name should retain its original value"
+            )
+            loaded = self.project.load(job.job_name)
+            self.assertFalse(
+                loaded._automatically_rename_on_save_using_input,
+                msg="The mangling preference should survive saving and loading"
+            )
+        finally:
+            job.remove()