1. Initial setup

Let the intensity of unpolarized light from the source reaching the screen without Polaroids be $I.$

Two Polaroids $P_1$ and $P_2$ are inserted, and the final intensity is given as $\frac{I}{2}.$

2. Use Malus' law

When unpolarized light passes through the first Polaroid, its intensity becomes half: $I_1 = \frac{I}{2}.$

If the angle between the transmission axes of $P_1$ and $P_2$ is $\theta$, then after passing through $P_2$: $I_2 = I_1 \cos^2\theta = \frac{I}{2}\cos^2\theta.$

Given that this equals $\frac{I}{2}$: $\frac{I}{2}\cos^2\theta = \frac{I}{2}.$

So, $\cos^2\theta = 1 \implies \theta = 0^\circ.$

Thus initially the two Polaroids are parallel.

3. After rotating $P_2$

Now let $P_2$ be rotated by an angle $\phi$. Then the angle between the axes becomes $\phi$, and the final intensity is $I' = \frac{I}{2}\cos^2\phi.$

We want $I' = \frac{3I}{8}.$

Therefore, $\frac{I}{2}\cos^2\phi = \frac{3I}{8}.$

Cancel $I$: $\frac{1}{2}\cos^2\phi = \frac{3}{8}.$

So, $\cos^2\phi = \frac{3}{4}.$

Hence, $\cos\phi = \frac{\sqrt{3}}{2}.$

Thus, $\phi = 30^\circ.$

4. Final answer

The required rotation is $\boxed{30}.$