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We show that there is only one embedding of $\mathrm B\mathrm{DI}_2$ in $\mathrm B\mathrm{F}_4$ at the prime $p=3$, up to self-maps of $\mathrm B\mathrm{DI}_2$. We also describe the effect of the group of self-equivalences of $\mathrm B\mathrm{F}_4$ at the prime $p=3$ on this embedding and then show that the Friedlander exceptional isogeny composed with a suitable Adams map is an involution of $\mathrm B\mathrm{F}_4$ whose homotopy fixed point set coincide with $\mathrm B\mathrm{DI}_2$ [ABSTRACT FROM AUTHOR]