  Two families of surfaces arise from considering cyclic branched covers of
$\mathbb{P}^{2}$ over smooth quartic curves. These consist of degree 2 del
Pezzo surfaces with a $\mathbb{Z}/2\mathbb{Z}$ action and $K3$ surfaces with a
$\mathbb{Z}/4\mathbb{Z}$ action. We compute the monodromy groups of both
families. In the first case, we obtain the Weyl group $W\left(E_{7}\right)$,
corresponding to the automorphisms of the $56$ lines contained in a degree $2$
del Pezzo surface. In the second case we obtain an arithmetic lattice: the
unitary group $U\left(h_{L_{-}}\right)$ of a type $\left(1, 6\right)$ quadratic
form over $\mathbb{Z}\left[i\right]$ by building on results of Kondo and
Allcock, Carlson, Toledo.
