A beam of light of intensity $I_0$ falls on a system of three polaroids which are arranged in succession…

For incident unpolarized light of intensity $I_0$, passage through the first polaroid halves the intensity: $I_1 = \frac{I_0}{2}$, resulting in light polarized along the first axis.

The second polaroid is rotated $60^\circ$ relative to the first. By Malus’s Law, the transmitted intensity is $I_2 = I_1 \cos^2 60^\circ = \frac{I_0}{2} \cdot \left(\frac{1}{2}\right)^2 = \frac{I_0}{8}$.

A third polaroid, rotated $60^\circ$ from the second, transmits $I_3 = I_2 \cos^2 60^\circ = \frac{I_0}{8} \cdot \frac{1}{4} = \frac{I_0}{32}$.

The fraction of incident intensity transmitted is $\frac{I_3}{I_0} = \frac{1}{32}$.

Final answer: $\boxed{\text{B}}$