Solution

Firstly we can say that moduls of friction forces is the same(force of connecting and matereals are same):

$$F_1 = F_2 = F$$

Whats more in the end:

$$\omega_1= \omega_3= -\omega_2$$

From second newton's law for rotation(for the first and second cylinders):

$$I\frac{\omega_1 - \omega}{t} = -F\cdot R$$

$$I\frac{\omega_2 - \omega}{t} = -2F\cdot R$$

Solving this system:

$$\omega_2 = 2\omega_1 - \omega$$

So we have:

$$\omega_1=\omega_3=\frac{\omega}{3}; \quad \omega_2=-\frac{\omega}{3}$$