Characters of some finite groups of Lie type with a restriction containing a linear character once

Let G be a finite group of characteristic p of one of the types symplectic group Sp(4,2f), Chevalley group G2(q), Suzuki group Sz(q) or Ree group Re(q). In this paper we show that if H is a Sylow p-subgroup of G then the restriction χH of χ to H has at least one linear constituent of multiplicity one for certain irreducible characters χ of G.