Higher mod $p$ Satake transforms: explicit formulas and images for $\mathrm{GL}_2$

Heyer's left adjoint to derived parabolic induction gives rise to higher mod $p$ Satake transforms \[ \mathcal S_V^i: \mathcal H_G(V) \longrightarrow \mathcal H_M\bigl(H^i(K_U,V)\bigr), \] whose extreme degrees recover Herzig's invariant and coinvariant transforms. For $G=\mathbf G(F)$, where $\math