diff --git a/Tests/test_imageops.py b/Tests/test_imageops.py
index 54f62d5c628..3a9ca3b4e1c 100644
--- a/Tests/test_imageops.py
+++ b/Tests/test_imageops.py
@@ -11,6 +11,10 @@
     hopper,
 )
 
+TYPE_CHECKING = False
+if TYPE_CHECKING:
+    from collections.abc import Callable
+
 
 class Deformer(ImageOps.SupportsGetMesh):
     def getmesh(
@@ -627,3 +631,37 @@ def test_autocontrast_preserve_one_color(color: tuple[int, int, int]) -> None:
         img, cutoff=10, preserve_tone=True
     )  # single color 10 cutoff
     assert_image_equal(img, out)
+
+
+@pytest.mark.parametrize(
+    "operation",
+    (
+        ImageOps.invert,
+        lambda im: ImageOps.posterize(im, 3),
+        ImageOps.solarize,
+    ),
+)
+def test_palette_operation(operation: Callable[[Image.Image], Image.Image]) -> None:
+    im = hopper("P")
+    result = operation(im)
+
+    assert result.mode == "P"
+    assert_image_equal(result.convert("RGB"), operation(im.convert("RGB")))
+
+
+def test_palette_operation_keeps_alpha() -> None:
+    im = Image.new("P", (1, 2))
+    im.putpalette((0, 0, 0, 0, 255, 0, 0, 128), "RGBA")
+    im.putpixel((0, 1), 1)
+
+    result = ImageOps.invert(im)
+
+    palette = result.getpalette("RGBA")
+    assert palette is not None
+    assert palette == [255, 255, 255, 0, 0, 255, 255, 128]
+
+
+def test_palette_histogram_operation_not_supported() -> None:
+    im = hopper("P")
+    with pytest.raises(NotImplementedError, match="mode P support coming soon"):
+        ImageOps.autocontrast(im)
diff --git a/src/PIL/ImageOps.py b/src/PIL/ImageOps.py
index 55d400e1d14..81b7021ca61 100644
--- a/src/PIL/ImageOps.py
+++ b/src/PIL/ImageOps.py
@@ -58,11 +58,28 @@ def _color(
     return color
 
 
-def _lut(image: Image.Image, lut: list[int]) -> Image.Image:
+def _lut(image: Image.Image, lut: list[int], palette: bool = False) -> Image.Image:
     if image.mode == "P":
-        # FIXME: apply to lookup table, not image data
-        msg = "mode P support coming soon"
-        raise NotImplementedError(msg)
+        if not palette:
+            # The lookup table was built from the palette indexes, not colors
+            msg = "mode P support coming soon"
+            raise NotImplementedError(msg)
+
+        # The lookup table can be applied to the palette colors
+        im = image.copy()
+        assert im.palette is not None
+        palette_mode = im.palette.mode
+        channels = len(palette_mode)
+        colors = im.getpalette(None)
+        assert colors is not None
+        im.putpalette(
+            [
+                v if palette_mode[i % channels] == "A" else lut[v]
+                for i, v in enumerate(colors)
+            ],
+            palette_mode,
+        )
+        return im
     elif image.mode in ("L", "RGB"):
         if image.mode == "RGB" and len(lut) == 256:
             lut = lut + lut + lut
@@ -655,7 +672,7 @@ def invert(image: Image.Image) -> Image.Image:
     :return: An image.
     """
     lut = list(range(255, -1, -1))
-    return image.point(lut) if image.mode == "1" else _lut(image, lut)
+    return image.point(lut) if image.mode == "1" else _lut(image, lut, palette=True)
 
 
 def mirror(image: Image.Image) -> Image.Image:
@@ -678,7 +695,7 @@ def posterize(image: Image.Image, bits: int) -> Image.Image:
     """
     mask = ~(2 ** (8 - bits) - 1)
     lut = [i & mask for i in range(256)]
-    return _lut(image, lut)
+    return _lut(image, lut, palette=True)
 
 
 def solarize(image: Image.Image, threshold: int = 128) -> Image.Image:
@@ -695,7 +712,7 @@ def solarize(image: Image.Image, threshold: int = 128) -> Image.Image:
             lut.append(i)
         else:
             lut.append(255 - i)
-    return _lut(image, lut)
+    return _lut(image, lut, palette=True)
 
 
 @overload