The second bounded cohomology of a free group of rank greater than 1 is infinite dimensional as a vector space over ℝ 4 . For a group G and its nth commutator subgroup G (n) , the quotient G/G (n) is amenable and the homomorphism induced from the inclusion homomorphism G (n)  → G is injective. In this article, we prove that if G (n) is free of rank greater than 1 for some finite ordinal n, then G is residually solvable and its second bounded cohomology is infinite dimensional. We prove its converse for a group generated by two elements. As for groups that are not residually solvable, we investigate the dimension of the second bounded cohomology of a perfect group. Also, some results on bounded cohomology of a connected CW complex X by applying a Quillen's plus construction X + to kill a perfect normal subgroup of π 1 X are given.