diff --git a/doc/source/whatsnew/v1.0.0.rst b/doc/source/whatsnew/v1.0.0.rst
index 2b147f948adb1..cbbbfff797ac4 100644
--- a/doc/source/whatsnew/v1.0.0.rst
+++ b/doc/source/whatsnew/v1.0.0.rst
@@ -284,6 +284,7 @@ Groupby/resample/rolling
 - Bug in :meth:`DataFrame.rolling` not allowing for rolling over datetimes when ``axis=1`` (:issue: `28192`)
 - Bug in :meth:`DataFrame.groupby` not offering selection by column name when ``axis=1`` (:issue:`27614`)
 - Bug in :meth:`DataFrameGroupby.agg` not able to use lambda function with named aggregation (:issue:`27519`)
+- Bug in :meth:`DataFrame.groupby` losing column name information when grouping by a categorical column (:issue:`28787`)
 
 Reshaping
 ^^^^^^^^^
diff --git a/pandas/core/groupby/grouper.py b/pandas/core/groupby/grouper.py
index 2d37121d28308..d7eaaca5ac83a 100644
--- a/pandas/core/groupby/grouper.py
+++ b/pandas/core/groupby/grouper.py
@@ -330,7 +330,8 @@ def __init__(
                 self._group_index = CategoricalIndex(
                     Categorical.from_codes(
                         codes=codes, categories=categories, ordered=self.grouper.ordered
-                    )
+                    ),
+                    name=self.name,
                 )
 
             # we are done
diff --git a/pandas/tests/groupby/test_categorical.py b/pandas/tests/groupby/test_categorical.py
index fcc0aa3b1c015..490ecaab03dab 100644
--- a/pandas/tests/groupby/test_categorical.py
+++ b/pandas/tests/groupby/test_categorical.py
@@ -674,7 +674,7 @@ def test_preserve_categories():
 
     # ordered=True
     df = DataFrame({"A": Categorical(list("ba"), categories=categories, ordered=True)})
-    index = CategoricalIndex(categories, categories, ordered=True)
+    index = CategoricalIndex(categories, categories, ordered=True, name="A")
     tm.assert_index_equal(
         df.groupby("A", sort=True, observed=False).first().index, index
     )
@@ -684,8 +684,8 @@ def test_preserve_categories():
 
     # ordered=False
     df = DataFrame({"A": Categorical(list("ba"), categories=categories, ordered=False)})
-    sort_index = CategoricalIndex(categories, categories, ordered=False)
-    nosort_index = CategoricalIndex(list("bac"), list("bac"), ordered=False)
+    sort_index = CategoricalIndex(categories, categories, ordered=False, name="A")
+    nosort_index = CategoricalIndex(list("bac"), list("bac"), ordered=False, name="A")
     tm.assert_index_equal(
         df.groupby("A", sort=True, observed=False).first().index, sort_index
     )
@@ -1193,3 +1193,17 @@ def test_groupby_categorical_axis_1(code):
     result = df.groupby(cat, axis=1).mean()
     expected = df.T.groupby(cat, axis=0).mean().T
     assert_frame_equal(result, expected)
+
+
+def test_groupby_cat_preserves_structure(observed):
+    # GH 28787
+    df = DataFrame([("Bob", 1), ("Greg", 2)], columns=["Name", "Item"])
+    expected = df.copy()
+
+    result = (
+        df.groupby("Name", observed=observed)
+        .agg(pd.DataFrame.sum, skipna=True)
+        .reset_index()
+    )
+
+    assert_frame_equal(result, expected)