If a spring is stretched does it retain the center of mass? If I stretch a spring, is the spring retaining its mass?
 A: Answer is yes, spring will accelerate with respect to you. As you hold spring in one hand, you will act as fix support to spring. Spring also have some mass so it will become spring-mass system (Assume mass of spring acting at end of spring, after calculation it will be mass of spring/3). After giving some initial displacement it will start vibrating (So free vibration case). Equation of motion will be $-k x = \frac{m}{3} a$ ➡️ So acceleration will be there but it is variable acceleration which is maximum at extreme end of vibration and zero at mean position.
A: It depends on whether or not your hand is moving with constant velocity.
Consider three portions of the spring, at the fixed end, in the middle, and at end you are holding.
The centre of mass of the spring is located at the middle of the spring.
If your hand is moving at a constant velocity of $2\,\rm cm/s$ relative to your eyes then the middle of the spring, and its centre of mass, will be moving at a constant velocity of $1\,\rm cm/s$.
However if you hand is accelerating so will the middle of the spring, and hence its centre of mass, be accelerating.
