Crossed homomorphisms from rank-2 abelian to exceptional p-groups

In this paper, we shall verify the conjecture Z1(A,G)≡0 (mod gcd(A/A′,G)) of [T. Asai, T. Yoshida, J. Algebra 160 (1993) 273–285] in the case where A is an abelian group of rank 2, and G is an exceptional p-group for a prime p, namely, one of the following groups: for p=2, a cyclic 2-group, a dihedral 2-group, a generalized quaternion 2-group, and a semidihedral 2-group; while for p>2, a cyclic p-group.