$\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$ is unacceptable

We classify the pairs of group morphisms $\mathrm{\Gamma }\to \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(7)$ which are element conjugate but not globally conjugate. As an application, we study the case where $\mathrm{\Gamma }$ is the Weil group of a $p$-adic local field, which is relevant to the recent approach to the local Langlands correspondence for ${\mathrm{G}}_2$ and ${\mathrm{P}\mathrm{G}\mathrm{S}\mathrm{p}}_6$ by Gan and Savin. As a second application, we improve some result of Kret and Shin about ${\mathrm{G}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}_7$-valued Galois representations.