What major challenges faced the federal government in reconstructing the South after the Civil War during the period from 1865 to 1877?

One of the major problems the federal government faced during Reconstruction was the disagreement between Radical Republicans in Congress, who wanted to pursue a far-reaching policy of Reconstruction, and President Johnson, who wanted a far more limited program. After he was impeached (and was not convicted), Johnson's power was limited.

The other problem the Radical Republicans faced was that the South wanted to return to their antebellum way of life. They were returning former Confederates to office, and they were also passing Black Codes to tie freed slaves to plantations and prevent their free movement. After Radical Republicans won the election of 1866, they pursued Military Reconstruction, which divided the South into military districts and required them to grant African-Americans citizenship (the 14th Amendment) and to give African-American men the vote (the 15th Amendment). By the late 1860s and early 1870s, the KKK was on the rise in many parts of the South, and there was violence to prevent these Amendments from being implemented. Jim Crow legislation, which instituted unequal treatment of African-Americans, became the law of the South. Eventually, the North withdrew troops from the South in 1877 without fully implementing laws that ensured that African-Americans could vote and own land. These reforms would unfortunately have to wait until the Civil Rights Movement of the 20th Century.