  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.
