Finite groups with non-trivial intersections of kernels of all but one irreducible characters

In this paper we consider finite groups G satisfying the followingý ýconditioný: ýG has two columns in its character table which differ by exactly oneý ýentryý. ýIt turns out that such groups exist and they are exactly the finite groupsý ýwith a non-trivial intersection of the kernels of all but one irreducibleý ýcharacters orý, ýequivalentlyý, ýfinite groups with an irreducible characterý ývanishing on all but two conjugacy classesý. ýWe investigate such groupsý ýand in particular we characterize their subclassý, ýwhich properly containsý ýall finite groups with non-linear characters of distinct degreesý, ýwhich were characterized by Berkovichý, ýChillag and Herzog in 1992ý.