In this paper, we determine the cokernel of the -th Johnson homomorphisms of the automorphism group of a free metabelian group for and . As a corollary, we obtain a lower bound on the rank of the graded quotient of the Johnson filtration of the automorphism group of a free group. Furthermore, by using the second Johnson homomorphism, we determine the image of the cup product map in the rational second cohomology group of the IA-automorphism group of a free metabelian group, and show that it is isomorphic to that of the IA-automorphism group of a free group which is already determined by Pettet. Finally, by considering the kernel of the Magnus representations of the automorphism group of a free group and a free metabelian group, we show that there are non-trivial rational second cohomology classes of the IA-automorphism group of a free metabelian group which are not in the image of the cup product map.