  We continue our study on the hypergeometric system $E(3,6)$ which describes
period integrals of the double cover family of K3 surfaces. Near certain
special boundary points in the moduli space of the K3 surfaces, we construct
the local solutions and determine the so-called mirror maps expressing them in
terms of genus two theta functions. These mirror maps are the K3 analogues of
the elliptic $\lambda$-function. We find that there are two non-isomorphic
definitions of the lambda functions corresponding to a flip in the moduli
space. We also discuss mirror symmetry for the double cover K3 surfaces and
their higher dimensional generalizations. A follow up paper will describe more
details of the latter.
