Inverse Satake isomorphism and change of weight

Let $G$ be any connected reductive $p$-adic group. Let $K\subset G$ be any special parahoric subgroup and $V,V'$ be any two irreducible smooth $\overline {\mathbb F}_p[K]$-modules. The main goal of this article is to compute the image of the Hecke bi-module $\operatorname{End}_{\overline {\mathbb F}_p[K]}(\operatorname{c-Ind}_K^G V, \operatorname{c-Ind}_K^G V')$ by the generalized Satake transform and to give an explicit formula for its inverse, using the pro-$p$ Iwahori Hecke algebra of $G$. This immediately implies the "change of weight theorem" in the proof of the classification of mod $p$ irreducible admissible representations of $G$ in terms of supersingular ones. A simpler proof of the change of weight theorem, not using the pro-$p$ Iwahori Hecke algebra or the Lusztig-Kato formula, is given when $G$ is split (and in the appendix when $G$ is quasi-split, for almost all $K$).