diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index 3cabb688..9d66e018 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -108,28 +108,35 @@ jobs:
       fail-fast: false
       matrix:
         python_version: ["3.7", "3.8", "3.9", "3.10", "3.11", "3.12"]
-        container_runtime: ["podman", "docker"]
-        update_runtime: [ true, false ]
+        container_runtime: ["podman"]
+        update_runtime: [true, false]
+        rootfull: [true, false]
 
-        exclude:
+        include:
           - container_runtime: "docker"
             python_version: "3.7"
-            update_runtime: true
+            update_runtime: false
+            rootfull: false
           - container_runtime: "docker"
             python_version: "3.8"
-            update_runtime: true
+            update_runtime: false
+            rootfull: false
           - container_runtime: "docker"
             python_version: "3.9"
-            update_runtime: true
+            update_runtime: false
+            rootfull: false
           - container_runtime: "docker"
             python_version: "3.10"
-            update_runtime: true
+            update_runtime: false
+            rootfull: false
           - container_runtime: "docker"
             python_version: "3.11"
-            update_runtime: true
+            update_runtime: false
+            rootfull: false
           - container_runtime: "docker"
             python_version: "3.12"
-            update_runtime: true
+            update_runtime: false
+            rootfull: false
 
     steps:
       - uses: actions/checkout@v4
@@ -143,6 +150,10 @@ jobs:
           key: nox-${{ matrix.python_version }}-${{ hashFiles('poetry.lock') }}
 
       - run: pip install --upgrade nox poetry nox-poetry
+        if: ${{ ! matrix.rootfull }}
+
+      - run: sudo pip install --upgrade nox poetry nox-poetry
+        if: ${{ matrix.rootfull }}
 
       - name: update the container runtime
         if: ${{ matrix.update_runtime }}
@@ -161,6 +172,7 @@ jobs:
       - run: |
           export CUR_USER="$(whoami)"
           sudo loginctl enable-linger ${CUR_USER}
+        if: ${{ ! matrix.rootfull }}
 
       - run: |
           mkdir ./tmp/
@@ -169,6 +181,16 @@ jobs:
           nox -s "test-${{ matrix.python_version }}(${{ matrix.container_runtime }})" -- -x -n auto --reruns 3 --pytest-container-log-level DEBUG
           nox -s "test-${{ matrix.python_version }}(${{ matrix.container_runtime }})" -- -x --reruns 3 --pytest-container-log-level DEBUG
           nox -s coverage
+        if: ${{ ! matrix.rootfull }}
+
+      - run: |
+          mkdir ./tmp/
+          chmod 1777 ./tmp
+          export TMPDIR="$(pwd)/tmp"
+          sudo --preserve-env nox -s "test-${{ matrix.python_version }}(${{ matrix.container_runtime }})" -- -x -n auto --reruns 3 --pytest-container-log-level DEBUG
+          sudo --preserve-env nox -s "test-${{ matrix.python_version }}(${{ matrix.container_runtime }})" -- -x --reruns 3 --pytest-container-log-level DEBUG
+          sudo --preserve-env nox -s coverage
+        if: ${{ matrix.rootfull }}
 
       - name: verify that no stray containers are left
         run: |
diff --git a/CHANGELOG.rst b/CHANGELOG.rst
index 54da14b7..a6369b0a 100644
--- a/CHANGELOG.rst
+++ b/CHANGELOG.rst
@@ -14,6 +14,10 @@ Breaking changes:
 
 Improvements and new features:
 
+- Add attributes :py:attr:`~pytest_container.inspect.ContainerInspect.name` and
+  :py:attr:`~pytest_container.inspect.ContainerNetworkSettings.ip_address`
+  exposing the container's name & IP
+
 - Add property
   :py:attr:`~pytest_container.container.ContainerBase.extra_entrypoint_args` to
   support appending arguments to the container launch command
diff --git a/pytest_container/inspect.py b/pytest_container/inspect.py
index f1185537..b71fd687 100644
--- a/pytest_container/inspect.py
+++ b/pytest_container/inspect.py
@@ -242,6 +242,9 @@ class ContainerNetworkSettings:
     #: list of ports forwarded from the container to the host
     ports: List[PortForwarding] = field(default_factory=list)
 
+    #: IP Address of the container, if it has one
+    ip_address: Optional[str] = None
+
 
 @dataclass(frozen=True)
 class Mount:
@@ -280,9 +283,12 @@ class ContainerInspect:
 
     """
 
-    #: The Container's ID
+    #: The container's ID
     id: str
 
+    #: the container's name
+    name: str
+
     #: program that has been launched inside the container
     path: str
 
diff --git a/pytest_container/runtime.py b/pytest_container/runtime.py
index 3b7fb7f5..20aace03 100644
--- a/pytest_container/runtime.py
+++ b/pytest_container/runtime.py
@@ -386,7 +386,11 @@ def _network_settings_from_inspect(
                         host_port=int(bindings[0]["HostPort"]),
                     )
                 )
-        return ContainerNetworkSettings(ports=ports)
+
+        net_settings = container_inspect["NetworkSettings"]
+        ip = net_settings.get("IPAddress") or None
+
+        return ContainerNetworkSettings(ports=ports, ip_address=ip)
 
     @staticmethod
     def _mounts_from_inspect(
@@ -517,6 +521,7 @@ def inspect_container(self, container_id: str) -> ContainerInspect:
         return ContainerInspect(
             config=conf,
             state=state,
+            name=inspect["Name"],
             id=inspect["Id"],
             path=inspect["Path"],
             args=inspect["Args"],
@@ -600,6 +605,8 @@ def inspect_container(self, container_id: str) -> ContainerInspect:
         return ContainerInspect(
             config=conf,
             state=state,
+            # docker prefixes the name with a / for reasons…
+            name=inspect["Name"].lstrip("/"),
             id=inspect["Id"],
             path=inspect["Path"],
             args=inspect["Args"],
diff --git a/tests/test_inspect.py b/tests/test_inspect.py
index 8d8c4eb6..cd5e5d02 100644
--- a/tests/test_inspect.py
+++ b/tests/test_inspect.py
@@ -7,14 +7,17 @@
 
 from .test_container_build import LEAP
 
+_CTR_NAME = "foobar-12345"
 
 IMAGE_WITH_EVERYTHING = DerivedContainer(
+    singleton=True,
+    extra_launch_args=["--name", _CTR_NAME],
     base=LEAP,
     containerfile="""VOLUME /src/
 EXPOSE 8080 666
 RUN useradd opensuse
 USER opensuse
-ENTRYPOINT /bin/false
+ENTRYPOINT /bin/bash
 ENV HOME=/src/
 ENV MY_VAR=
 ENV SUFFIX_NAME=dc=example,dc=com
@@ -23,13 +26,18 @@
 )
 
 
-@pytest.mark.parametrize("container", [IMAGE_WITH_EVERYTHING], indirect=True)
-def test_inspect(container: ContainerData, container_runtime: OciRuntimeBase):
-    inspect = container.inspect
+@pytest.mark.parametrize(
+    "container_per_test", [IMAGE_WITH_EVERYTHING], indirect=True
+)
+def test_inspect(
+    container_per_test: ContainerData, container_runtime: OciRuntimeBase, host
+) -> None:
+    inspect = container_per_test.inspect
 
-    assert inspect.id == container.container_id
+    assert inspect.id == container_per_test.container_id
+    assert inspect.name == _CTR_NAME
     assert inspect.config.user == "opensuse"
-    assert inspect.config.entrypoint == ["/bin/sh", "-c", "/bin/false"]
+    assert inspect.config.entrypoint == ["/bin/sh", "-c", "/bin/bash"]
 
     assert (
         "HOME" in inspect.config.env and inspect.config.env["HOME"] == "/src/"
@@ -39,9 +47,9 @@ def test_inspect(container: ContainerData, container_runtime: OciRuntimeBase):
     # prefixes it with `localhost` and the full build tag
     # (i.e. `pytest_container:$digest`), while docker just uses the digest
     expected_img = (
-        str(container.container)
+        str(container_per_test.container)
         if container_runtime.runner_binary == "docker"
-        else f"localhost/pytest_container:{container.container}"
+        else f"localhost/pytest_container:{container_per_test.container}"
     )
 
     assert inspect.config.image == expected_img
@@ -59,3 +67,7 @@ def test_inspect(container: ContainerData, container_runtime: OciRuntimeBase):
         and isinstance(inspect.mounts[0], VolumeMount)
         and inspect.mounts[0].destination == "/src"
     )
+
+    assert inspect.network.ip_address or "" == host.check_output(
+        f'{container_runtime.runner_binary} inspect --format "{{{{ .NetworkSettings.IPAddress }}}}" {_CTR_NAME}'
+    )