On tame automorphisms of some metabelian groups (Q1586970)

Let \(M\) be a variety of groups and let \(F_r(M)\) be the free group of rank \(r\) of this variety. Recall that an automorphism \(\alpha\) of \(F_r(M)\) is tame if there exists an automorphism \(\overline\alpha\) of the free group \(F_r\) of rank \(r\) such that \(\alpha\) is induced by \(\overline\alpha\). Otherwise \(\alpha\) is said to be wild. The author finds necessary and sufficient conditions for an automorphism \(\alpha\) to be tame in the case when \(M\) is a variety of metabelian groups of nilpotency class \(m\) (\(m\geq 3\)). The author also proves that the following problem has an algorithmic solution: Is an automorphism \(\alpha\) of \(F_r(M)\) tame?