Notion of algebra subgroup may differ for different interpretations of the same group as an algebra group

Statement
Suppose $$q$$ is a prime power. It is possible to construct an algebra group $$G$$ over the field of $$q$$ elements and two non-isomorphic nilpotent associative algebras $$N_1,N_2$$ both having algebra group $$G$$ with the property that there is a subgroup $$H$$ of $$G$$ that is an algebra subgroup with respect to the algebra group structure from $$N_1$$ but not with respect to the algebra group structure from $$N_2$$.

Case of the prime two
In this case, let $$G$$ be direct product of Z4 and Z2 and let $$N_1,N_2$$ be the nilpotent associative algebras described on the page algebra group structures for direct product of Z4 and Z2. Let $$H$$ be either copy of Z4 in direct product of Z4 and Z2. $$H$$ is a subgroup with the algebra group structure arising from $$N_1$$, but not the one from $$N_2$$.