Does $2X\neq X$ imply $2Y\neq Y$?

Question

Let $n$ be a sufficiently large integer and $X\subseteq \mathbf{Z}_n$ a finite set of residues modulo $n$ such that $0\in X$ and $2X \neq X$. Fix also $y \in 2X\setminus X$ and define $Y=\{x \in X:y\notin X+x\}$.

Question. Is it true that $2Y \neq Y$?

Ps. Here $A+B:=\{a+b:a \in A,b \in B\}$, $2A:=A+A$, and $A+b:=A+\{b\}$.

Answer 1

No, the simplest choice for $X$ works; take $n=3$ and $X=\{0,1\}$. Then \begin{align} 2X&=\{0,1,2\}\\ 2X\backslash X&=\{2\}\\ y&=2\\ Y&=\{0\}\\ 2Y&=\{0\}=Y. \end{align}