Abstract The Johnson homomorphisms τ k (k⩾1) give Abelian quotients of a series of certain subgroups of the mapping class group of a surface. Morita determined the rational image of the second Johnson homomorphism τ 2. In this paper, we study the structure of the torsion part of the cokernel of τ 2. First, we determine the rank of the cokernel over Z 2 . Although we do it first by computing explicitly, later we improve the proof, using the Birman–Craggs homomorphism, obtained by the classical Rohlin invariant of homology 3-spheres. Since τ 2 is equivariant with respect to the action of the mapping class group, Im τ 2 is Sp(2g; Z) -invariant and hence Sp(2g; Z) acts on the cokernel. Moreover, computing this action explicitly, we show that the action reduces to that of the finite symplectic group Sp(2g; Z 2) .