Large orbits of odd-order subgroups of solvable linear groups

Suppose that G is a finite solvable group and V is a finite, faithful and completely reducible G-module. Let H be an odd-order subgroup of G, then H has at least two regular orbits on V⊕V. Suppose in addition that |V| is odd, then there exists v∈V in a regular orbit of F(G)∩H such that CH(v)⊆F2(G). Let G be a solvable group, H be an odd-order Hall π-subgroup of G, V be a faithful G-module, over possibly different finite fields of odd π-characteristic and assume that VOπ(G) is completely reducible, then there exists v∈V such that CH(v)⊆Oπ(G).