Reduction type of genus-3 curves in a special stratum of their moduli space

We study a 3-dimensional stratum ${\mathcal{M}}_{3,V}$ of the moduli space ${\mathcal{M}}_3$ of curves of genus $3$ parameterizing curves $Y$ that admit a certain action of $V=C_2\times C_2$. We determine the possible types of the stable reduction of these curves to characteristic different from $2$. We define invariants for ${\mathcal{M}}_{3,V}$ and characterize the occurrence of each of the reduction types in terms of them. We also calculate the $j$-invariant (resp. the Igusa invariants) of the irreducible components of positive genus of the stable reduction of $Y$ in terms of the invariants.