Non proper HNN extensions and uniform uniform exponential growth

If a finitely generated torsion free group K has the property that all finitely generated subgroups S of K are either small or have growth constant bounded uniformly away from 1 then a non proper HNN extension G of K, that is a semidirect product of K by the integers, has the same property. Here small means cyclic or, if the automorphism has no periodic conjugacy classes, free abelian of bounded rank.