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In "Curves on Heisenberg invariant quartic surfaces in projective 3-space",Eklund showed that a general $(\mathbb{Z}/2\mathbb{Z})^{4}$-invariant quarticK3 surface contains at least $320$ conics. In this paper we analyse the fieldof definition of those conics as well as their Monodromy group. As a result, weprove that the moduli space $(\mathbb{Z}/2\mathbb{Z})^{4}$-invariant quartic K3surface with a marked conic has $10$ irreducible components.

On the Monodromy and Galois Group of Conics Lying on Heisenberg Invariant Quartic K3 Surfaces