Notes on automorphisms of surfaces of general type with $p_g=0$ and $K^2=7$

Let $S$ be a smooth minimal complex surface of general type with $p_g=0$ and $K^2=7$. We prove that any involution on $S$ is in the center of the automorphism group of $S$. As an application, we show that the automorphism group of an Inoue surface with $K^2=7$ is isomorphic to $\mathbb{Z}_2^2$ or $\mathbb{Z}_2 \times \mathbb{Z}_4$. We construct a $2$-dimensional family of Inoue surfaces with automorphism groups isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_4$.