The nonvanishing first Hochschild cohomology of twisted finite simple group algebras

Let $G$ be a finite simple group and $k$ be an algebraically closed field of prime characteristic dividing the order of $G$. We show that for all $2$-cocycles $\alpha \in Z^2(G;k^{\times })$, the first Hochschild cohomology group of the twisted group algebra $H{H}^1(k_{\alpha }G)$ is nonzero.