Embeddings of ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(q)$ in exceptional groups of Lie type over a field of characteristic $\mathrm{\ne }2,3$

Let $\mathit{G}$ be an algebraic group of exceptional Lie type in characteristic $p$, $G={\mathit{G}}^{\sigma }$ its fixed-point subgroup under the action of a Steinberg endomorphism $\sigma$, and $\bar{G}$ an almost simple group with socle $G$. A maximal subgroup $M<\bar{G}$ is called non-generic if it is almost simple and its socle is not isomorphic to a group of Lie type in characteristic $p$. A finite subgroup $H$ of $\mathit{G}$ is Lie primitive if it does not lie in any proper closed positive-dimensional subgroup of $\mathit{G}$; it is Lie imprimitive if it lies in a positive-dimensional subgroup $\mathit{X}$ of $\mathit{G}$; it is strongly imprimitive if $\mathit{X}$ can be chosen to be stable under the action of $N_{\mathrm{A}\mathrm{u}\mathrm{t}\mathit{G}}(H)$, where $\mathrm{A}\mathrm{u}\mathrm{t}\mathit{G}$ is the group generated by inner, diagonal, graph, and field automorphisms of $\mathit{G}$. We study the possible embeddings of a non-generic primitive simple group $H$ in the adjoint algebraic group $\mathit{G}$ in characteristic coprime to $\mathrm{\mid }H\mathrm{\mid }$ when $(\mathit{G},H)$ is one of $({\mathit{F}}_4,{\mathrm{P}\mathrm{S}\mathrm{L}}_2(25))$, $({\mathit{F}}_4,{\mathrm{P}\mathrm{S}\mathrm{L}}_2(27))$, $({\mathit{E}}_7,{\mathrm{P}\mathrm{S}\mathrm{L}}_2(29))$, $({\mathit{E}}_7,{\mathrm{P}\mathrm{S}\mathrm{L}}_2(37))$. In particular, we construct copies of $H$ in $G$ over a suitable finite field $k$, and use them to deduce the number of conjugacy classes of $H$ in $G$ and $\mathit{G}$, and whether $N_{\bar{G}}(H)$ is a maximal subgroup of $\bar{G}$. We also study the case of $H\simeq {\mathrm{A}\mathrm{l}\mathrm{t}}_6\simeq {\mathrm{P}\mathrm{S}\mathrm{L}}_2(9)$ when $\mathit{G}$ is one of ${\mathit{F}}_4$ and ${\mathit{E}}_6$ in characteristic coprime to $\mathrm{\mid }H\mathrm{\mid }$, and show that in such cases $H$ is a strongly imprimitive subgroup of $\mathit{G}$; in particular, $N_{\bar{G}}(H)$ is not a maximal subgroup of $\bar{G}$.