This paper is organized as follows. In Section 2, we use work of Sebbar \[Se\] to determine the groups $\Gamma$ corresponding to case (ii) of Theorem 2. These are subgroups of $PSL(2,{{\mathbb Z}})$, and associated to each $PSL(2,{{\mathbb Z}})$-conjugacy class there is a natural elliptic fibration over the base curve, defined over ${{\mathbb Q}}$. The total spaces of these fibrations are elliptic modular surfaces in the sense of Shioda \[Sh1\]. Moreover each is an extremal K3 surface (namely their Picard number is 20, the maximum possible). We explain the relation between the motive of their transcendental cycles, and specific CM forms of weight $3$ using a result of Livné on orthogonal rank 2 motives in \[L2\].

Section 3 contains our main results: we construct our examples, verify the required properties, and analyze the interesting part of their cohomology. (See the final Remarks 8 (2) for the other parts.) Then in Section 4 we give explicit formulas for the weight $3$ cusp forms and defining equations for the elliptic fibrations of Section 2.

The paper is supplemented by the article of Hulek and Verrill \[HV\] where they treat Kummar varieties, one of which is the case associated to the modular group 