A student steps onto a stationary elevator and stands on a bathroom scale. The elevator then travels from the top of the building to the bottom. The student records the reading on the scale as a function of time.

At what time(s) does the student have maximum downward velocity?

$\textbf{(A)}$ At all times between $2 s$ and $4 s$
$\textbf{(B)}$ At $4 s$ only
$\textbf{(C)}$ At all times between $4 s$ and $22 s$
$\textbf{(D)}$ At $22 s$ only
$\textbf{(E)}$ At all times between $22 s$ and $24 s$
1. **Understanding the problem**: The student is in an elevator that starts from rest, accelerates, travels at a constant speed, and then decelerates to a stop. The student records the reading on the scale, which changes due to the acceleration and deceleration of the elevator.

2. **Identifying the periods of acceleration and deceleration**: 
   - The elevator accelerates from rest between $2$ seconds and $4$ seconds.
   - The elevator decelerates to a stop between $22$ seconds and $24$ seconds.

3. **Determining the maximum downward velocity**:
   - The elevator reaches its maximum downward velocity at the end of the acceleration period, which is at $4$ seconds.
   - The elevator maintains this maximum downward velocity until it starts decelerating at $22$ seconds.

4. **Conclusion**: The elevator has its maximum downward velocity during the period when it is traveling at a constant speed, which is between $4$ seconds and $22$ seconds.

Thus, the correct answer is $\boxed{\textbf{(C)}}$.