Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ (in the sense that $Y$ is hyperelliptic and the morphism $X\to Y$ is finite (not necessarily etale of degree two)) with the property that $\# \mathrm{Aut}(Y) > \# \mathrm{Aut}(X)$.

The answer is yes when asked like this; take $Y$ to be $\mathbb P^1$ and $X\to Y$ the hyperelliptic map. (This is under the pretense that $\mathbb P^1$ is also a hyperelliptic curve.)

But what if we also ask $Y$ to be of genus at least two? That is:

Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ with the properties that $$g(Y)\geq 2, \ \textrm{and} \quad \# \mathrm{Aut}(Y) > \# \mathrm{Aut}(X).$$

I emphasize that the map $X\to Y$ is only assumed to be finite in this question (it is not necessarily of degree two, or etale)