Let K K be a number field, let A A be a finite-dimensional semisimple K K -algebra, and let Λ \Lambda be an O K \mathcal {O}_{K} -order in A A . It was shown in previous work that, under certain hypotheses on A A , there exists an algorithm that for a given (left) Λ \Lambda -lattice X X either computes a free basis of X X over Λ \Lambda or shows that X X is not free over Λ \Lambda . In the present article, we generalize this by showing that, under weaker hypotheses on A A , there exists an algorithm that for two given Λ \Lambda -lattices X X and Y Y either computes an isomorphism X → Y X \rightarrow Y or determines that X X and Y Y are not isomorphic. The algorithm is implemented in Magma for A = Q [ G ] A=\mathbb {Q}[G] , Λ = Z [ G ] \Lambda =\mathbb {Z}[G] , and Λ \Lambda -lattices X X and Y Y contained in Q [ G ] \mathbb {Q}[G] , where G G is a finite group satisfying certain hypotheses. This is used to investigate the Galois module structure of rings of integers and ambiguous ideals of tamely ramified Galois extensions of Q \mathbb {Q} with Galois group isomorphic to Q 8 × C 2 Q_{8} \times C_{2} , the direct product of the quaternion group of order 8 8 and the cyclic group of order 2 2 .