Let $R$ be a finite local $\mathbb{F}_q$-algebra. Assume that $R\cong R^*$ as left $R$-modules. Are there any known results about the abelianization $(R^{\times})_{ab}$?

(We can factor $R$ be the ideal $I$ generated by all additive commutators $xy-yx$. We then get a map $(R^{\times})_{ab}\to (R/I)^{\times}$ but this map does not have to be an isomorphism).