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We consider the deformation spaces of some singular product-quotient surfaces$X=(C_1 \times C_2)/G$, 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 ofTodorov surfaces with $p_g=1$, $q=0$ and $2\le K^2\le 8$ by using$\mathbb{Q}$-Gorenstein smoothings.

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