Regular orbits of nilpotent subgroups of solvable linear groups

We settle a conjecture of Walter Carlip (1994) [2, Conjecture 1.3]. Suppose that G is a finite solvable group, V is a finite faithful FG-module over a field of characteristic p and assume Op(G)=1. Let H be a nilpotent subgroup of G. Assume that H involves no wreath product Zr≀Zr for r=2 or r a Mersenne prime, then H has at least one regular orbit on V.