Finite groups in which every irreducible character vanishes on at most two conjugacy classes

It is known that if G is a finite non-abelian group in which every irreducible character vanishes on at most one conjugacy class, then G is a Frobenius group with a Frobenius complement of order 2 and Frobenius kernel of odd order. In particular G is solvable. This paper studies the class of finite groups G in which every irreducible character vanishes on at most two conjugacy classes. There are such nonsolvable groups. We show that A5 and PSL(2, 7) are the only nonsolvable groups in this class. We also show that each solvable group in this class is either a certain type of a Frobenius group, or is very close to being one.