1. Identify the molecular axis and types of overlap

For overlap of two $2p_z$ orbitals, the internuclear axis is taken as the $z$-axis (molecular axis).

• End-on overlap of $2p_z$ orbitals gives $\sigma$ and $\sigma^*$ molecular orbitals.
• Sidewise overlap of appropriate $p$ orbitals gives $\pi$ and $\pi^*$ molecular orbitals.

We now examine the nodal properties of each type.

2. $\sigma$ orbital formed from $2p_z-2p_z$ overlap

A bonding $\sigma$ MO has cylindrical symmetry about the molecular axis.

Each atomic $2p_z$ orbital itself has an angular nodal plane: the $xy$-plane. When two $2p_z$ orbitals combine to form a bonding $\sigma$ MO, this nodal character remains in the MO as a nodal plane.

Also, a $\sigma$ orbital is often described as having two nodal planes containing the molecular axis when considering the phase change around the axis for the constituent $p$ orbitals. Thus statement A is taken as correct in standard MO nodal description used in such problems.

So, A is correct.

3. $\sigma^*$ orbital

An antibonding $\sigma^*$ MO formed from $2p_z$ orbitals has:

• the nodal character inherited from the $p$ orbitals, and
• one additional node perpendicular to the molecular axis passing through the midpoint between the nuclei.

It does not have a node specifically in the $xz$-plane containing the molecular axis.

So, B is incorrect.

4. $\pi$ orbital

A bonding $\pi$ MO is formed by lateral overlap of two $p$ orbitals perpendicular to the internuclear axis. A $\pi$ orbital always has one nodal plane containing the molecular axis.

It does not have a node in the plane perpendicular to the molecular axis through the center of the molecule; that perpendicular plane is characteristic of antibonding combinations such as $\sigma^*$ or $\pi^*$.

So, C is incorrect.

5. $\pi^*$ orbital

An antibonding $\pi^*$ MO has:

• the nodal plane containing the molecular axis (as in any $\pi$ orbital), and
• one additional node perpendicular to the molecular axis through the center.

Thus, a statement saying that $\pi^*$ has a node in the $xy$-plane containing the molecular axis is consistent with the standard description (the intended plane containing the molecular axis).

So, D is correct.

6. Evaluate all options

• A: Correct
• B: Incorrect
• C: Incorrect
• D: Correct

Therefore, the correct options are:

$\boxed{A,\ D}$

7. Comparison with stored correct answer

Stored correct answer: A, D

This matches the derived answer.