If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

Question

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribution as $aX$ for some real $a$, what are the possible characteristic functions of $X$?

Let $\varphi_X(t)$ be characteristic function of $X$. Then $$\varphi_X^2(t)=\varphi_{X_1+X_2}(t)=\varphi_{aX}(t)=\varphi_X(at).$$

By iterating the above functional equation we get $$\varphi_X^{2n}(t)=\varphi_X(a^nt)$$ for $n\in\mathbb{N}$. I have no idea how to approach the problem form here. Any ideas?