Deformations of product-quotient surfaces and reconstruction of Todorov surfaces via $\mathbb{Q}$-Gorenstein smoothing

We consider the deformation spaces of some singular product-quotient surfaces $X=(C_1\times C_2)\mathrm{/}G$, where the curves $C_i$ have genus 3 and the group $G$ is isomorphic to ${\mathbb{Z}}_4$. As a by-product, we give a new construction of Todorov surfaces with $p_g=1$, $q=0$ and $2\le K^2\le 8$ by using $\mathbb{Q}$-Gorenstein smoothings.