  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 $\mathbb{Z}_2 \times
\mathbb{Z}_4$.
