Final answer:
The Milky Way is moving away from most other galaxies due to cosmic expansion, except for a few nearby ones like Andromeda. The question proposes a theoretical scenario involving relativistic speeds and intergalactic travel, which is not currently practical.
Step-by-step explanation:
The statement is true: the Milky Way Galaxy is moving further away from most other galaxies due to the expansion of the universe. This phenomenon is evidenced by observations that galaxies are flying apart from one another, with more distant galaxies moving away faster, which is consistent with the theory of the Big Bang. The Andromeda Galaxy, which is the closest large galaxy to the Milky Way and is 2 million light years away, is one of the few exceptions, as it is actually moving towards the Milky Way.
When considering sending an exploratory probe to a galaxy receding from us at 0.900c, where c is the speed of light, one must factor in both the velocity of the galaxy and the desired approach velocity. This is a highly theoretical scenario because with current technology, reaching such relativistic speeds is not feasible. Moreover, the answer to the question about the required probe velocity and travel time involves complex calculations based on the special theory of relativity.
The time it would take for a radio signal to return would simply be the time it takes for the light to travel back the same distance under constant velocity assumptions. However, practical intergalactic travel and communication remain beyond our current capabilities.