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Let $\boldsymbol{G}$ be an algebraic group of exceptional Lie type incharacteristic $p$, $G=\boldsymbol{G}^{\sigma}$ its fixed-point subgroup underthe action of a Steinberg endomorphism $\sigma$, and $\overline{G}$ an almostsimple group with socle $G$. A maximal subgroup $M<\overline{G}$ is callednon-generic if it is almost simple and its socle is not isomorphic to a groupof Lie type in characteristic $p$. A finite subgroup $H$ of $\boldsymbol{G}$ isLie primitive if it does not lie in any proper closed positive-dimensionalsubgroup of $\boldsymbol{G}$; it is Lie imprimitive if it lies in apositive-dimensional subgroup $\boldsymbol{X}$ of $\boldsymbol{G}$; it isstrongly imprimitive if $\boldsymbol{X}$ can be chosen to be stable under theaction of $N_{\mathrm{Aut}\boldsymbol{G}}(H)$, where$\mathrm{Aut}\boldsymbol{G}$ is the group generated by inner, diagonal, graph,and field automorphisms of $\boldsymbol{G}$.  We study the possible embeddings of a non-generic primitive simple group $H$in the adjoint algebraic group $\boldsymbol{G}$ in characteristic coprime to$|H|$ when $(\boldsymbol{G},H)$ is one of$(\boldsymbol{F}_4,\mathrm{PSL}_2(25))$,$(\boldsymbol{F}_4,\mathrm{PSL}_2(27))$,$(\boldsymbol{E}_7,\mathrm{PSL}_2(29))$,$(\boldsymbol{E}_7,\mathrm{PSL}_2(37))$. In particular, we construct copies of$H$ in $G$ over a suitable finite field $k$, and use them to deduce the numberof conjugacy classes of $H$ in $G$ and $\boldsymbol{G}$, and whether$N_{\overline{G}}(H)$ is a maximal subgroup of $\overline{G}$. We also studythe case of $H\simeq\mathrm{Alt}_6\simeq\mathrm{PSL}_2(9)$ when$\boldsymbol{G}$ is one of $\boldsymbol{F}_4$ and $\boldsymbol{E}_6$ incharacteristic coprime to $|H|$, and show that in such cases $H$ is a stronglyimprimitive subgroup of $\boldsymbol{G}$; in particular, $N_{\overline{G}}(H)$is not a maximal subgroup of $\overline{G}$.

Embeddings of $\mathrm{PSL}_2(q)$ in exceptional groups of Lie type over a field of characteristic $\ne2,3$