As Will Sawin suggests, the answer to the last version of the question is negative. If we tak $G$ to be a non-Abelian finite simple group which is not a doubly transitive permutation group, and $\chi$ to be a non-trivial complex irreducible character of last degree o $G,$ then it is impossible to write $chi$ + any multiple of teh trivial character as a sum of characters each induced from linear characters of (not necssarily proper) subgroups.