Let $A$ be a semi-simple finite dimensional algebra over complex filed $C$, and let $V$ be a direct sum of two isomorphic simple $A$-modules. Find the automorphism group of the $A$-module $V$.
My thoughts: based on the definition of semi-simple finite algebra, we can know that $A$ can be represented by $A\cong A_1\oplus A_2$, where $A_i$ is simple finite algebra. Then, we feel a little confused how to analyze this question, can someone help me?
Here is the definition of semisimple module:https://en.wikipedia.org/wiki/Semisimple_module