diff --git a/pandas/core/groupby/generic.py b/pandas/core/groupby/generic.py
index 8cd727e744519d..8e53972c952759 100644
--- a/pandas/core/groupby/generic.py
+++ b/pandas/core/groupby/generic.py
@@ -970,6 +970,11 @@ 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]]
+                if obj.shape[1] == 1:
+                    # Avoid call to self.values that can occur in DataFrame
+                    #  reductions; see GH#28949
+                    obj = obj.iloc[:, 0]
+
                 s = groupby(obj, self.grouper)
                 try:
                     result = s.aggregate(lambda x: alt(x, axis=self.axis))
@@ -978,17 +983,29 @@ def _cython_agg_blocks(self, how, alt=None, numeric_only=True, min_count=-1):
                     # continue and exclude the block
                     deleted_items.append(locs)
                     continue
+
+                # unwrap DataFrame to get array
+                assert len(result._data.blocks) == 1
+                result = result._data.blocks[0].values
+                if result.ndim == 1 and isinstance(result, np.ndarray):
+                    result = result.reshape(1, -1)
+
             finally:
+                assert not isinstance(result, DataFrame)
+
                 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)
 
-                    if result.ndim == 1 and isinstance(result, np.ndarray):
+                    if block.is_extension and isinstance(result, np.ndarray):
                         # e.g. block.values was an IntegerArray
+                        # (1, N) case can occur if block.values was Categorical
+                        #  and result is ndarray[object]
+                        assert result.ndim == 1 or result.shape[0] == 1
                         try:
                             # Cast back if feasible
                             result = type(block.values)._from_sequence(
-                                result, dtype=block.values.dtype
+                                result.ravel(), dtype=block.values.dtype
                             )
                         except ValueError:
                             # reshape to be valid for non-Extension Block
diff --git a/pandas/core/groupby/groupby.py b/pandas/core/groupby/groupby.py
index 6f2868482b7986..fa651794698403 100644
--- a/pandas/core/groupby/groupby.py
+++ b/pandas/core/groupby/groupby.py
@@ -1344,13 +1344,14 @@ def f(self, **kwargs):
                     raise
                 except DataError:
                     pass
-                except Exception:
-                    # TODO: the remaining test cases that get here are from:
-                    #  - AttributeError from _cython_agg_blocks bug passing
-                    #    DataFrame to make_block; see  GH#28275
-                    #  - TypeError in _cython_operation calling ensure_float64
-                    #    on object array containing complex numbers;
-                    #    see test_groupby_complex, test_max_nan_bug
+                except (TypeError, NotImplementedError):
+                    # TODO:
+                    #  - TypeError: this is reached via test_groupby_complex
+                    #    and can be fixed by implementing _group_add for
+                    #    complex dtypes
+                    #  - NotImplementedError: reached in test_max_nan_bug,
+                    #    raised in _get_cython_function and should probably
+                    #    be handled inside _cython_agg_blocks
                     pass
 
                 # apply a non-cython aggregation