Let the circuit be the standard bridge-type arrangement: left branch has battery $V_1$ in series with $R_1$, right branch has battery $V_2$ in series with $R_3$, and the resistor $R_2$ connects the midpoints.
For current through $R_2$ to be zero, the two junctions connected by $R_2$ must be at the same potential.

We solve using the balance condition.

1. Condition for zero current through $R_2$

If current through $R_2$ is zero, then no current flows across the bridge branch, so the two side branches behave independently.

Let the top and bottom nodes be common. Then current in left branch is $I_1 = \frac{V_1}{R_1}$ and current in right branch is $I_3 = \frac{V_2}{R_3}.$

Potential drop from the top node to the midpoint on the left is $I_1R_1$ and on the right is $I_3R_3$. For the midpoint potentials to be equal, we require the fractional division to match. Equivalently, the bridge balance condition is $\frac{V_1}{V_2} = \frac{R_1}{R_3}.$

Now check the options.

2. Check option A

Given: $V_1=V_2, \qquad R_1=R_2=R_3.$ Then $\frac{V_1}{V_2}=1, \qquad \frac{R_1}{R_3}=1.$ Condition is satisfied.

So A is correct.

3. Check option B

Given: $V_1=V_2, \qquad R_1=2R_2=R_3.$ Thus $R_1=R_3.$ Hence $\frac{V_1}{V_2}=1, \qquad \frac{R_1}{R_3}=1.$ Condition is satisfied.

So B is correct.

4. Check option C

Given: $V_1=2V_2, \qquad 2R_1=2R_2=R_3.$ From $2R_1=R_3$, we get $R_3=2R_1 \Rightarrow \frac{R_1}{R_3}=\frac{1}{2}.$ But $\frac{V_1}{V_2}=2.$ These are not equal, so the balance condition is not satisfied.

So C is incorrect.

5. Check option D

Given: $2V_1=V_2 \Rightarrow \frac{V_1}{V_2}=\frac{1}{2}.$ Also, $2R_1=R_2=R_3 \Rightarrow R_3=2R_1 \Rightarrow \frac{R_1}{R_3}=\frac{1}{2}.$ Condition is satisfied.

So D is correct.

6. Final answer

The current through $R_2$ is zero for: $\boxed{A,\ B,\ D}$

This matches the stored correct answer.