This was Lincoln's original plan, but since he died, it never happened.
In the late 1900s, Liberia had political instability, since Americo-Liberians (American slave descendants) were 3% of the population, but held all the power.
In 1865, the population of Liberia was about 720,000, of whom 20,000 were Americo-Liberians. There were approximately 4.5 million African-Americans.
A couple thoughts
1. Some would surely die on the boat ride to Liberia, and some would surely find some way to stay in the US, though if the government went all in, this would be less than 100,000 people. Liberia would get 4.3 million African-Americans, at most.
2. Liberia would now have 5.7 million people, of whom about 5 million (about seven-eights) are Americo-Liberian. Americo-Liberians would probably be able to maintain their power indefinitely, the same way Turkey rules over its Kurdish minority (~15% of the population) and Egypt rules over its Coptic minority (~10% of the population).
3. Liberia today has between 4-5 million people. Could 1865 Liberia even support 5-6 million inhabitants?
4. America would now be 99% White. This probably butterflies Jim Crow away, since one of the biggest reasons for it was that the White southern leaders didn't want the south become mostly mulatto. African-Americans were about half of the population in many former Confederate states. I wonder if America would still have been open to the massive turn of the 20th century wave of immigration. On one hand, they would need the extra labor because 4.5 million Black people leaving would put a huge dent in the labor force. On the other hand, with virtually no Black population, there would be less of a motivation for the government to try and attract White European immigrants to increase the White share of the population.
5. I am virtually certain the Hart-Celler act would have never passed, since its proponents latched on to the African-American Civil Rights Movement to get it through.
What I think this would lead to:
1. A USA that is at least 95% White today
2. A Liberia that is about 7/8 America-Liberian, and still run by them
3. Ethnic and religious differences between White Americans would be more salient, since Anti-Black racism is a big part of what encouraged Anglo Whites to accept Ethnic Whites.
4. America would not be as powerful, economically or militarily.
5. America would probably have less social tension, since a lot of it is race-based, though the tension between different White groups might just be greater instead.
6. America would be more agrarian