On simple-minded systems over representation-finite self-injective algebras

Let $A$ be a representation-finite self-injective algebra over an algebraically closed field $k$. We give a new characterization for an orthogonal system in the stable module category $A$-$\text{stmod}$ to be a simple-minded system. As a by-product, we show that every Nakayama-stable orthogonal system in $A$-$\text{stmod}$ extends to a simple-minded system.