Let $A$ be a commutative Frobenius algebra over a field $K$ (we can assume that $A$ is local).

Let $B=\{v_i \}$ be a vector space basis of $A$ containing the unit of $A$. Let $M_i:=v_i A$ and $M:= \bigoplus_{}^{}{M_i}$ and $C:=\underline{End_A}(M)$ the stable endomorphism ring of $M$.

Question: Is $C$ independent of the choosen basis $B$ up to isomorphism?