  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 of
Todorov surfaces with $p_g=1$, $q=0$ and $2\le K^2\le 8$ by using
$\mathbb{Q}$-Gorenstein smoothings.
