Conjecture on the alternating or symmetric action of the Markoff automorphism group

Let $H(p)$ be the permutation group induced by the action of $\operatorname{Out}(\mathbf{F}_{2})$ on $\mathbb{Y}_{-2}(\mathbb{F}_{p})$, and let

$$n=|\mathbb{Y}_{-2}(\mathbb{F}_{p})|.$$

Here $\mathbb{Y}_{-2}(\mathbb{F}_{p})$ is the set of $N$-orbits in $\mathbb{X}_{-2}^{*}(\mathbb{F}_{p})$, where $N$ is the Klein four-group of even sign changes. Alternating-group conjecture. For every prime $p>3$, $H(p)$ is the full alternating group or the full symmetric group according to the congruence class of $p$ modulo $16$; specifically, the preceding theorem shows that $H(p)\leq A_n$ exactly when $p\equiv3\pmod{16}$, and the conjecture asserts the corresponding complete determination of $H(p)$. This conjecture is motivated by computations for $p\leq47$ and by the transitive action of $\operatorname{Out}(\mathbf{F}_{2})$ on the relevant Markoff surface orbits; the exact group identification beyond the parity criterion remains open.