Genuine $C_n$-equivariant $\mathrm{T}\mathrm{M}\mathrm{F}$

We determine the $\mathrm{T}\mathrm{M}\mathrm{F}$-module structures of the genuine $C_2$-equivariant $\mathrm{T}\mathrm{M}\mathrm{F}$ with $\mathrm{R}\mathrm{O}(C_2)$-gradings and of the $C_3$-equivariant $\mathrm{T}\mathrm{M}\mathrm{F}$. Moreover, we propose a general strategy for studying $C_n$-equivariant $\mathrm{T}\mathrm{M}\mathrm{F}$ via $U(1)$-equivariant $\mathrm{T}\mathrm{M}\mathrm{F}$ and a duality phenomenon in equivariant $\mathrm{T}\mathrm{M}\mathrm{F}$.