Let $G$ be a locally profinite group and $K$ an open compact subgroup (mod the center), then Bushnell has shown that the following three statements are equivalent for a finite dimensional representation $\rho$ of $K$:

the compact induction $ind_K^G \rho$ is admissible

compact induction is isomorphic to the usual (analytic) induction $Ind_K^G \rho$

the compact induction decomposes into a finite sum of irreducible representations

My main question is the following:

What are necessary and sufficient condition on $\rho$ for $ind_K^G \rho$ to be admissible?

I am mostly interested in the case, where $K$ maximal compact subgroup (mod the center) in $GL(2,F)$ for a non archimedean local field $F$.

I am little bit familiar with Bushnell-Henniart's monograph on Langlands for $GL(2)$ and also much with the work of Kutzko and others, and understand the classification of supercuspidal representation of $GL(2,F)$. So, I am aware of some sufficient conditions for those representation arising from cuspidal type strata.

I also am aware of the basic strategy for proving admissibility with the restriction-induction formula and Frobenius reciprocity by proving finite-dimensionality of the algebra $$ Hom_G( ind_K^G \rho, ind_K^G \rho) = \bigoplus_{\gamma \in G//K} Hom_{K \cap K^\gamma}( \rho^\gamma, \rho),$$ and using Iwahori decomposition to spell out $K \cap K^\gamma$, but I do not see what to require about $\rho$ to make this work. So I hope somebody cant hint me to the relevant section in the literature, or give a hint.

Is is necessary and sufficient that $Res_{N \cap K} \rho$ does not contain the trivial representation, for any unipotent group $N \subset G$?

On a related matter, consider the distinct two maximal compact groups (mod the center) $K$ and $K'$ with $\rho \in Rep(K)$ and $\rho' \in Rep(K')$ such that the compact induction is admissible.

Is it true that $$ Hom_G( ind_K^G \rho, ind_{K'}^G \rho') = \bigoplus_{\gamma \in K' \backslash G/K} Hom_{K' \cap K^\gamma} ( \rho^\gamma, \rho) = 0?$$

Thanks a lot.