Some permutations over ${\mathbb F}_p$ concerning primitive roots

Let $p$ be an odd prime and let ${\mathbb F}_p$ denote the finite field with $p$ elements. Suppose that $g$ is a primitive root of ${\mathbb F}_p$. Define the permutation $\tau_g:\,{\mathcal H}_p\to{\mathcal H}_p$ by $$\tau_g(b):=\begin{cases} g^b,&\text{if }g^b\in{\mathcal H}_p,\\ -g^b,&\text{if }g^b\not\in{\mathcal H}_p,\\ \end{cases}$$ for each $b\in{\mathcal H}_p$, where ${\mathcal H}_p=\{1,2,\ldots,(p-1)/2\}$ is viewed as a subset of ${\mathbb F}_p$. In this paper, we investigate the sign of $\tau_g$. For example, if $p\equiv 5\pmod{8}$, then $$(-1)^{|\tau_g|}=(-1)^{\frac{1}{4}(h(-4p)+2)}$$ for every primitive root $g$, where $h(-4p)$ is the class number of the imaginary quadratic field ${\mathbb Q}(\sqrt{-4p})$.