Monodromy of Kodaira Fibrations of Genus $3$

A Kodaira fibration is a non-isotrivial fibration $f\text{\hspace{0.17em}⁣}:S\to B$ from a smooth algebraic surface $S$ to a smooth algebraic curve $B$ such that all fibers are smooth algebraic curves of genus $g$. Such fibrations arise as complete curves inside the moduli space ${\mathcal{M}}_g$ of genus $g$ algebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case $g=3$ and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of ${\mathcal{M}}_3$ parametrizing curves whose Jacobians have extra endomorphisms.