Let A and B be ordinary 2-isogenous elliptic curves defined over a finite field F of odd characteristic. Suppose the groups A(F) and B(F) are isomorphic. We determine necessary and sufficient conditions for the groups A(L) and B(L) to be isomorphic for all finite extensions L/F. This complements recent work in which we considered the similar question for l-isogenous curves, when l is odd.

We thank the anonymous referee for a careful reading of the draft and detailed comments which improved the exposition and content of the paper.