diff --git a/test/test_transforms.py b/test/test_transforms.py
index ec413b2b34c..6a57d4faa1e 100644
--- a/test/test_transforms.py
+++ b/test/test_transforms.py
@@ -689,6 +689,85 @@ def test_single_trans_env_check(self, out_keys):
         )
         check_env_specs(env)
 
+    @pytest.mark.parametrize("cat_dim", [-1, -2, -3])
+    @pytest.mark.parametrize("cat_N", [3, 10])
+    @pytest.mark.parametrize("device", get_default_devices())
+    def test_with_permute_no_env(self, cat_dim, cat_N, device):
+        torch.manual_seed(cat_dim * cat_N)
+        pixels = torch.randn(8, 5, 3, 10, 4, device=device)
+
+        a = TensorDict(
+            {
+                "pixels": pixels,
+            },
+            [
+                pixels.shape[0],
+            ],
+            device=device,
+        )
+
+        t0 = Compose(
+            CatFrames(N=cat_N, dim=cat_dim),
+        )
+
+        def get_rand_perm(ndim):
+            cat_dim_perm = cat_dim
+            # Ensure that the permutation moves the cat_dim
+            while cat_dim_perm == cat_dim:
+                perm_pos = torch.randperm(ndim)
+                perm = perm_pos - ndim
+                cat_dim_perm = (perm == cat_dim).nonzero().item() - ndim
+                perm_inv = perm_pos.argsort() - ndim
+            return perm.tolist(), perm_inv.tolist(), cat_dim_perm
+
+        perm, perm_inv, cat_dim_perm = get_rand_perm(pixels.dim() - 1)
+
+        t1 = Compose(
+            PermuteTransform(perm, in_keys=["pixels"]),
+            CatFrames(N=cat_N, dim=cat_dim_perm),
+            PermuteTransform(perm_inv, in_keys=["pixels"]),
+        )
+
+        b = t0._call(a.clone())
+        c = t1._call(a.clone())
+        assert (b == c).all()
+
+    @pytest.mark.skipif(not _has_gym, reason="Test executed on gym")
+    @pytest.mark.parametrize("cat_dim", [-1, -2])
+    def test_with_permute_env(self, cat_dim):
+        env0 = TransformedEnv(
+            GymEnv("Pendulum-v1"),
+            Compose(
+                UnsqueezeTransform(-1, in_keys=["observation"]),
+                CatFrames(N=4, dim=cat_dim, in_keys=["observation"]),
+            ),
+        )
+
+        env1 = TransformedEnv(
+            GymEnv("Pendulum-v1"),
+            Compose(
+                UnsqueezeTransform(-1, in_keys=["observation"]),
+                PermuteTransform((-1, -2), in_keys=["observation"]),
+                CatFrames(N=4, dim=-3 - cat_dim, in_keys=["observation"]),
+                PermuteTransform((-1, -2), in_keys=["observation"]),
+            ),
+        )
+
+        torch.manual_seed(0)
+        env0.set_seed(0)
+        td0 = env0.reset()
+
+        torch.manual_seed(0)
+        env1.set_seed(0)
+        td1 = env1.reset()
+
+        assert (td0 == td1).all()
+
+        td0 = env0.step(td0.update(env0.full_action_spec.rand()))
+        td1 = env0.step(td0.update(env1.full_action_spec.rand()))
+
+        assert (td0 == td1).all()
+
     def test_serial_trans_env_check(self):
         env = SerialEnv(
             2,