Non-isomorphic nilpotent associative algebras may have isomorphic algebra groups

Statement

Let $q$ be any prime power. It is possible to construct associative algebras $N_{1}$ and $N_{2}$ over $\mathbb {F} _{q}$, both of which are nilpotent (i.e., all products of a certain length or more are zero), such that $N_{1}$ and $N_{2}$ are not isomorphic as $\mathbb {F} _{q}$-algebras, but the algebra group corresponding to $N_{1}$ is isomorphic to the algebra group corresponding to $N_{2}$.

Proof

Currently, we only have the proof for the prime two, but a similar proof technique should work in other cases.

Proof for the prime two

See direct product of Z4 and Z2 is an algebra group for two non-isomorphic nilpotent associative algebras.