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In [] we studied the nonclassical Hopf-Galois module structure of rings of algebraic integers in some tamely ramified extensions of local and global fields, and proved a partial generalisation of Noether’s theorem to this setting. In this paper we consider tame Galois extensions of number fields $L/K$ with group $G\cong C_2\times C_2$ and study in detail the local and global structure of the ring of integers ${𝔒}_L$ as a module over its associated order ${𝔄}_H$ in each of the Hopf algebras $H$ giving a nonclassical Hopf-Galois structure...