diff --git a/pandas/core/groupby/generic.py b/pandas/core/groupby/generic.py
index c0436e9389078..08bbd3940d4c2 100644
--- a/pandas/core/groupby/generic.py
+++ b/pandas/core/groupby/generic.py
@@ -30,6 +30,7 @@
     ensure_int64,
     ensure_platform_int,
     is_bool,
+    is_categorical_dtype,
     is_datetimelike,
     is_dict_like,
     is_integer_dtype,
@@ -48,12 +49,7 @@
 from pandas.core.frame import DataFrame
 from pandas.core.generic import ABCDataFrame, ABCSeries, NDFrame, _shared_docs
 from pandas.core.groupby import base
-from pandas.core.groupby.groupby import (
-    GroupBy,
-    _apply_docs,
-    _transform_template,
-    groupby,
-)
+from pandas.core.groupby.groupby import GroupBy, _apply_docs, _transform_template
 from pandas.core.index import Index, MultiIndex, _all_indexes_same
 import pandas.core.indexes.base as ibase
 from pandas.core.internals import BlockManager, make_block
@@ -161,10 +157,15 @@ def _cython_agg_blocks(self, how, alt=None, numeric_only=True, min_count=-1):
                 result, _ = self.grouper.aggregate(
                     block.values, how, axis=agg_axis, min_count=min_count
                 )
-            except NotImplementedError:
+            except NotImplementedError as err:
                 # generally if we have numeric_only=False
                 # and non-applicable functions
                 # try to python agg
+                if "type does not support" in str(err):
+                    # exception raised by NumPy, not pandas
+                    # e.g. "timedelta64 type does not support prod operations"
+                    deleted_items.append(locs)
+                    continue
 
                 if alt is None:
                     # we cannot perform the operation
@@ -174,7 +175,7 @@ def _cython_agg_blocks(self, how, alt=None, numeric_only=True, min_count=-1):
 
                 # call our grouper again with only this block
                 obj = self.obj[data.items[locs]]
-                s = groupby(obj, self.grouper)
+                s = obj.groupby(self.grouper)
                 try:
                     result = s.aggregate(lambda x: alt(x, axis=self.axis))
                 except TypeError:
@@ -182,10 +183,28 @@ def _cython_agg_blocks(self, how, alt=None, numeric_only=True, min_count=-1):
                     # continue and exclude the block
                     deleted_items.append(locs)
                     continue
+
+                if is_object_dtype(block.dtype) and how in ["prod", "cumprod", "sum"]:
+                    # s.aggregate is not reliable for e.g. `prod` with strings
+                    result = no_result
+                    # TODO: why are we raising here and continuing elsewhere?
+                    # (tests.groupby.test_function.test_arg_passthru breaks
+                    #  if we continue here)
+                    raise
+
+                if is_categorical_dtype(block.dtype):
+                    # restore Categorical; not all dtypes are conserved by agg
+                    # TODO: will this be right for e.g. sum?
+                    result = result.astype(block.dtype)
+
+                assert len(result._data.blocks) == 1
+                result = result._data.blocks[0].values
+
             finally:
                 if result is not no_result:
                     # see if we can cast the block back to the original dtype
                     result = maybe_downcast_numeric(result, block.dtype)
+                    assert not isinstance(result, DataFrame)
                     newb = block.make_block(result)
 
             new_items.append(locs)
diff --git a/pandas/tests/groupby/test_function.py b/pandas/tests/groupby/test_function.py
index d89233f2fd603..c27af160c9898 100644
--- a/pandas/tests/groupby/test_function.py
+++ b/pandas/tests/groupby/test_function.py
@@ -171,13 +171,12 @@ def test_arg_passthru():
         result = f(numeric_only=False)
         tm.assert_frame_equal(result.reindex_like(expected), expected)
 
-    # TODO: min, max *should* handle
-    # categorical (ordered) dtype
     expected_columns = Index(
         [
             "int",
             "float",
             "string",
+            "category_string",
             "category_int",
             "datetime",
             "datetimetz",
@@ -212,7 +211,9 @@ def test_arg_passthru():
         result = f(numeric_only=False)
         tm.assert_index_equal(result.columns, expected_columns)
 
-    expected_columns = Index(["int", "float", "string", "category_int", "timedelta"])
+    expected_columns = Index(
+        ["int", "float", "string", "category_string", "category_int", "timedelta"]
+    )
     for attr in ["sum"]:
         f = getattr(df.groupby("group"), attr)
         result = f()