Crossed homomorphisms from rank-2 Abelian to exceptional \(p\)-groups. (Q1419009)

Let \(A\) and \(G\) be finite groups with group homomorphism \(\varphi\colon A\to\Aut(G)\). A map \(\xi\colon A\to G\) is said to be a crossed homomorphism if \(\xi(ab)=\xi(a)\varphi(a)(\xi(b))\). Let \(Z^1(A,G)\) be the set of all crossed homomorphisms from \(A\) to \(G\). \textit{T. Asai} and \textit{T. Yoshida} [J. Algebra 160, No. 1, 273-285 (1993; Zbl 0827.20034)] conjectured that \(|Z^1(A,G)|\equiv 0\pmod{\gcd(|A/[A,A]|,|G|)}\). The authors confirm this conjecture for Abelian groups \(A\) of rank 2 and a group \(G\) which is either a cyclic \(p\)-group or a 2-group of maximal class.