diff --git a/doc/source/whatsnew/v1.3.0.rst b/doc/source/whatsnew/v1.3.0.rst
index 6a85bfd852e19..a249d0cf39a0b 100644
--- a/doc/source/whatsnew/v1.3.0.rst
+++ b/doc/source/whatsnew/v1.3.0.rst
@@ -339,6 +339,7 @@ Groupby/resample/rolling
 - Fixed bug in :meth:`DataFrameGroupBy.sum` and :meth:`SeriesGroupBy.sum` causing loss of precision through using Kahan summation (:issue:`38778`)
 - Fixed bug in :meth:`DataFrameGroupBy.cumsum`, :meth:`SeriesGroupBy.cumsum`, :meth:`DataFrameGroupBy.mean` and :meth:`SeriesGroupBy.mean` causing loss of precision through using Kahan summation (:issue:`38934`)
 - Bug in :meth:`.Resampler.aggregate` and :meth:`DataFrame.transform` raising ``TypeError`` instead of ``SpecificationError`` when missing keys had mixed dtypes (:issue:`39025`)
+- Bug in :meth:`.DataFrameGroupBy.idxmin` and :meth:`.DataFrameGroupBy.idxmax` with ``ExtensionDtype`` columns (:issue:`38733`)
 
 Reshaping
 ^^^^^^^^^
diff --git a/pandas/core/arrays/base.py b/pandas/core/arrays/base.py
index 9a8b37e0785e0..b0979218e099c 100644
--- a/pandas/core/arrays/base.py
+++ b/pandas/core/arrays/base.py
@@ -30,7 +30,7 @@
 from pandas.compat.numpy import function as nv
 from pandas.errors import AbstractMethodError
 from pandas.util._decorators import Appender, Substitution
-from pandas.util._validators import validate_fillna_kwargs
+from pandas.util._validators import validate_bool_kwarg, validate_fillna_kwargs
 
 from pandas.core.dtypes.cast import maybe_cast_to_extension_array
 from pandas.core.dtypes.common import (
@@ -596,13 +596,17 @@ def argsort(
             mask=np.asarray(self.isna()),
         )
 
-    def argmin(self):
+    def argmin(self, skipna: bool = True) -> int:
         """
         Return the index of minimum value.
 
         In case of multiple occurrences of the minimum value, the index
         corresponding to the first occurrence is returned.
 
+        Parameters
+        ----------
+        skipna : bool, default True
+
         Returns
         -------
         int
@@ -611,15 +615,22 @@ def argmin(self):
         --------
         ExtensionArray.argmax
         """
+        validate_bool_kwarg(skipna, "skipna")
+        if not skipna and self.isna().any():
+            raise NotImplementedError
         return nargminmax(self, "argmin")
 
-    def argmax(self):
+    def argmax(self, skipna: bool = True) -> int:
         """
         Return the index of maximum value.
 
         In case of multiple occurrences of the maximum value, the index
         corresponding to the first occurrence is returned.
 
+        Parameters
+        ----------
+        skipna : bool, default True
+
         Returns
         -------
         int
@@ -628,6 +639,9 @@ def argmax(self):
         --------
         ExtensionArray.argmin
         """
+        validate_bool_kwarg(skipna, "skipna")
+        if not skipna and self.isna().any():
+            raise NotImplementedError
         return nargminmax(self, "argmax")
 
     def fillna(self, value=None, method=None, limit=None):
diff --git a/pandas/tests/extension/base/methods.py b/pandas/tests/extension/base/methods.py
index 7e7f1f1a6e025..3518f3b29e8c2 100644
--- a/pandas/tests/extension/base/methods.py
+++ b/pandas/tests/extension/base/methods.py
@@ -128,6 +128,16 @@ def test_argreduce_series(
         result = getattr(ser, op_name)(skipna=skipna)
         tm.assert_almost_equal(result, expected)
 
+    def test_argmax_argmin_no_skipna_notimplemented(self, data_missing_for_sorting):
+        # GH#38733
+        data = data_missing_for_sorting
+
+        with pytest.raises(NotImplementedError, match=""):
+            data.argmin(skipna=False)
+
+        with pytest.raises(NotImplementedError, match=""):
+            data.argmax(skipna=False)
+
     @pytest.mark.parametrize(
         "na_position, expected",
         [
diff --git a/pandas/tests/groupby/test_function.py b/pandas/tests/groupby/test_function.py
index 8d7fcbfcfe694..f532e496ccca9 100644
--- a/pandas/tests/groupby/test_function.py
+++ b/pandas/tests/groupby/test_function.py
@@ -531,10 +531,16 @@ def test_idxmin_idxmax_returns_int_types(func, values):
         }
     )
     df["c_date"] = pd.to_datetime(df["c_date"])
+    df["c_date_tz"] = df["c_date"].dt.tz_localize("US/Pacific")
+    df["c_timedelta"] = df["c_date"] - df["c_date"].iloc[0]
+    df["c_period"] = df["c_date"].dt.to_period("W")
 
     result = getattr(df.groupby("name"), func)()
 
     expected = DataFrame(values, index=Index(["A", "B"], name="name"))
+    expected["c_date_tz"] = expected["c_date"]
+    expected["c_timedelta"] = expected["c_date"]
+    expected["c_period"] = expected["c_date"]
 
     tm.assert_frame_equal(result, expected)