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.