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I can give only an answer to the 2nd question:

The functor $\mathcal{F}$ is a composition of restriction to $G(o)$ and then projection onto the subrepresentation of $G(o)$, which have kernel $\bmod p$.

Both of these functors are very wellbehaved by the Peter Weyl Theorem.

So all the properties you want follow, but it seems to me that you will study only very few representation with this (finitely many?).

I can give only an answer to the 2nd question:

The functor $\mathcal{F}$ is a composition of restriction to $G(o)$ and then projection onto the subrepresentation of $G(o)$, which have kernel $\bmod p$.

Both of these functors are very wellbehaved by the Peter Weyl Theorem.

So all the properties you want follow in the case of a reductive group over a local non archimedean field, but it seems to me that you will study only very few representation with this (finitely many?).

Edit: The observation with cuspidal mapsto supercuspidal and parabolic induced to parabolic induced is best explained with the restriction induction formula for composing restriction and induction, perhaps modulo the assumption that supercuspidal are induced from maximal compacts (I do not know, if this is true in the generality you want this.)

I can give only an answer to the 2nd question:

All the functor $\mathcal{F}$ is first restriction to $G(o)$ and then projection onto the subrepresentation of $GL_2(o)$, which have kernel $\Gamma(p)$.

Both of these Functors are very wellbehaved by the Peter Weyl Theorem.

I can give only an answer to the 2nd question:

The functor $\mathcal{F}$ is a composition of restriction to $G(o)$ and then projection onto the subrepresentation of $G(o)$, which have kernel $\bmod p$.

Both of these functors are very wellbehaved by the Peter Weyl Theorem.

So all the properties you want follow, but it seems to me that you will study only very few representation with this (finitely many?).

I can give only apartial answer to the 2nd question, but it will be too long for a comment:

For the parabolic induction:

The Iwasawa decomposition gives you

$$ Res_{G(o)} Ind_{P(F)}^{G(F)} \chi \cong Ind_{P(o)}^{G(O)} Res_{P(o)} \chi $$

Then you find that

$$ \left( Ind_{P(o)}^{G(o)} Res_{P(o)} \chi \right)^{\Gamma(p)} = Ind_{P(f)}^{G(f)} \tilde{\chi} $$

where $f$ is the residue field. Where $\tilde{\chi}$ is the irreducible sub representation of $\chi$ by the $M(p)$ invariant vectors for the Levi subgroup $M$ of $P$.

For the observation with supercuspidal, I have only a comment:

I assume that the super cuspidal is induced from an open compact group $G(o)$, thenone can use Frobenius reciprocity, i.e. consider a representation of $\chi \in G(o/p)$

$$ Hom_{G(o/p)}(\chi, \left( Res_{G(o)} Ind_{G(o)}^G \pi\right)^{\Gamma(p)}) = \bigoplus_{m \in G //G(o)} Hom_G ( \chi, Ind_{G(o/p) \cap G(o/p)^m}^{G(o/p)} \pi )$$ with the Cartan decomposition can be computed quite fast.

For $GL(n)$, you can see for which representation the induced one is actually irreducible and supercuspidal (see e.g. Kutzko and Bushnell).

For exactness:

Restriction to open compact subgroup is always an exact functor, since the representation theory of compact groups is completely irreducible.

I can give only an answer to the 2nd question:

All the functor $\mathcal{F}$ is first restriction to $G(o)$ and then projection onto the subrepresentation of $GL_2(o)$, which have kernel $\Gamma(p)$.

Both of these Functors are very wellbehaved by the Peter Weyl Theorem.