Take $G = S_3 \times S_4$ and consider the unique two-dimensional irreducible representation of $S_3$ and the unique two-dimensional irreducible representation of $S_4$. These have the same character values $2,0,-1$ (in fact they are both pullbacks of the same representation under two different homomorphisms $S_4 \to S_3$ ) but are not related by any combination of the operations.
Galois conjugation is irrelevant since the characters are rational. Twisting by one-dimensional characters can only twist by quadratic characters and doesn't affect the value on order $3$ elements - i.e. $3$-cycles in $S_3$, $3$-cycles in $S_4$, or products of $3$-cycles in both $S_3$ and $S_4$. So it remains to consider outer automorphisms, which would have to permute these three conjugacy classes. But no outer automorphism can since their centralizers have different orders: $3 \cdot 24, 6 \cdot 3,$ and $3 \cdot 3$ respectively.