Mixed quasi-\'etale surfaces, new surfaces of general type with $p_g=0$ and their fundamental group

We call a projective surface $X$ mixed quasi-'etale quotient if there exists a curve $C$ of genus $g(C)\ge 2$ and a finite group $G$ that acts on $C\times C$exchangingthefactorssuchthat$X=(C\times C)/G$andthemap$C\times C \rightarrow X$ has finite branch locus. The minimal resolution of its singularities is called mixed quasi-'etale surface. We study the mixed quasi-'etale surfaces under the assumption that $(C\times C)\mathrm{/}{G}^0$ has only nodes as singularities, where $G^0◃G$ is the index two subgroup of the elements that do not exchange the factors. We classify the minimal regular surfaces with $p_g=0$ whose canonical model is a mixed quasi-'etale quotient as above. All these surfaces are of general type and as an important byproduct, we provide an example of a numerical Campedelli surface with topological fundamental group ${\text{bbZ}}_4$, and we realize 2 new topological types of surfaces of general type. Three of the families we construct are $\text{bbQ}$-homology projective planes.