$\DeclareMathOperator{\cd}{cd}$If $N$ is normal subgroup of $G$, is it true that $$|\cd(N)|\leqslant|\cd(G)|?$$ Here $\cd(G)$ denote the set of all distinct irreducible characters degree. If it is true , how can I prove it? What happens when $N=O_p(G)$?
I am using Clifford theory to produce irreducible characters of $G$ associated with every irreducible character of $N$ with distinct degree, but I can't prove that they have distinct degrees.
Your question has a negative answer in general. Note that $|\mathrm{cd}(A_9)|=16$ and $|\mathrm{cd}(S_9)|=15$. This is unlikely to be the smallest example, but it is the smallest alternating example.