We have found that Co-2 × 2 islands grown on an Ag/Ge(111)-√3 × √3 surface have hcp structure with the (11-20) orientation. The island evolution involves transformation of the unit cell shape from parallelogram into rectangular, which is accompanied by the island shape transformation from hexagonal into stripe-like. Identified are two crystallographic directions for the island growth, the pseudo- and the pseudo-[1-100]. We have observed the occurrence of a lateral shift between the topmost and the underlying bilayers in the case of the island growth along the pseudo- direction. In contrast, the topmost and the underlying bilayers are unshifted for the growth along the pseudo-[1-100] direction.