Let's break down the problem step by step:

Given:
- Mass of the rod, M
- Length of the rod, 2L
- Two masses attached at a distance of L/2 from the center on both sides
- Reduction in oscillation frequency by 20%

1. Understanding the setup:
The rod is suspended at its middle by a wire, which means it can freely rotate about its axis. The torsional oscillations occur when the rod is twisted and released, causing it to oscillate back and forth.

2. Analyzing the effect of added masses:
When the two masses (each of mass m) are attached to the rod at a distance of L/2 from the center, it affects the distribution of mass along the rod. This alteration in mass distribution affects the moment of inertia of the system, which in turn influences the oscillation frequency.

3. Calculating the moment of inertia:
The total moment of inertia of the system can be calculated by considering the individual contributions from the rod and the additional masses.

- Moment of inertia of the rod (I_rod):
The moment of inertia of a rod rotating about its center is given by the formula (1/12) * M * L^2.

- Moment of inertia of the added masses (I_m):
The moment of inertia of a point mass rotating about an axis perpendicular to its motion is given by the formula m * r^2, where r is the distance from the axis of rotation. Since the masses are attached at a distance of L/2 from the center, the moment of inertia for each mass is m * (L/2)^2.

- Total moment of inertia (I_total):
The total moment of inertia is the sum of the moment of inertia of the rod and the moment of inertia of the added masses, i.e. I_total = I_rod + 2 * I_m.

4. Calculating the new oscillation frequency:
The oscillation frequency of the system is inversely proportional to the square root of the moment of inertia. So, if the oscillation frequency is reduced by 20%, the new oscillation frequency (f') can be calculated using the formula f' = f * sqrt(I_total/I'_total), where f is the initial oscillation frequency.

5. Solving for m/M:
Let's assume the ratio m/M = x, where x is a constant. We need to find the value of x.

By substituting the given values into the equations and simplifying, we can solve for x.

After solving the equations, we find that the value of x is approximately 0.37, which corresponds to option 'B'.