I know a lot of places where the following is sparsely proved, but I remember there was some paper where I read it in basically the same form I write it, but unfortunately I can't remember where it was.


Let $\mathcal{H}$ be the Hecke algebra of a reductive $p$-adic group, and $K$ be a compact open subgroup of $G$. There is a restriction functor $r : \mathcal{M}(\mathcal{H})\rightarrow \mathcal{M}(\mathcal{H}\_K)$ defined by $r(V)=e\_K V=V^K$ having the following properties.


* $r$ has a left adjoint, the *induction functor* $i: \mathcal{M}(\mathcal{H}\_K)\rightarrow \mathcal{M}(\mathcal{H})$ (*Frobenius reciprocity*).
* $i$ carries admissible representations to admissible representations.
* The pair $(r,i)$ gives a bijection between $\mathcal{M}(\mathcal{H}\_K)$ and the subcategory of $\mathcal{M}(\mathcal{H})$ of representations that are generated by $K$-fixed vectors.



Basic theorem on induction for representations of $p$-adic groups
The general setting for your question is the theory of types as developped by Bushnell and Kutzko:


Smooth representations of reductive p-adic groups: structure theory via types. Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634.


First in order that your question make sense, let us clarify a few things.


$\bullet$ The good "induction functor" here is 
$$
i(M) = {\mathcal H}(G)\otimes\_{{\mathcal H}(K)} M
$$
where $M$ is a left ${\mathcal H}(K)$-module. Here you consider ${\mathcal H}(G)$ as a right ${\mathcal H}(K)$-module and left $G$-module. 


$\bullet$ **It is false** that $i$ takes "admissible modules" to "admissible modules". For instance if $M={\mathbb C}$ is the "trivial" character of ${\mathcal H}(K)$, then $i(M)$ is not admissible! Indeed if $L$ is a compact open subgroup of $G$, $i(M)^L$ is the space of $(L,K)$⁻bi-invariant functions on $G$ with finite support modulo $(L,K)$, an infinite dimensional space.


$\bullet$ You refer the pair $(r,i)$ as a "bijection". Of course you mean an equivalence of categories. **This is false in general!**


In order to get an equivalence of categories you need strong assumptions on $K$. As Peter McNamara writes, you need that $K$ have some nice "Iwahori decompositions". Typically a maximal compact subgroup of $G$ does not fulfill this condition. However this is true for a general reductive group $G$ and:


$\bullet$ $K=I$ an Iwahori subgroup of $G$ (Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup. Invent. Math. 35 (1976), 233–259). 


$\bullet$ $K$ is a certain "congruence subgroup" ("le centre de Bernstein" in Deligne, P.; Kazhdan, D.; Vignéras, M.-F. Représentations des algèbres centrales simples p-adiques. (French) [Representations of central simple p-adic algebras] Representations of reductive groups over a local field, 33–117, Travaux en Cours, Hermann, Paris, 1984. 


The aim of Bushnell and Kutzko's theory is to generalize this as follows. Instead of taking the trivial character of $K$, you fix an irreducible smooth representation $\rho$. You define an idempotent $e\_\rho$ in ${\mathcal H}(G)$ such that $e\_\rho V$ is the $\rho$-isotypic component $V^\rho$ of $V$, for all smooth $G$-modules $V$. You define the Hecke algebra of the pair $(K,\rho )$ by 
${\mathcal H}\_\rho = e\_\rho \star {\mathcal H}(G)\star e\_\rho$. You have two functors between the category of $G$-modules $V$ generated by $V^\rho$ and the category of left ${\mathcal H}\_\rho$-modules: $r(V)=V^\rho$, and $i(M) = {\mathcal H}(G)\otimes\_{{\mathcal H}\_\rho} M$. Then by definition $(K,\rho )$ is a type in $G$ if these functors define an equivalence of categories. 


Given an irreducible representation $\pi$ of $G$, exhibiting a nice type $(K,\rho )$ such that $\pi\_{\mid K}$ contains $\rho$ gives a lot of informations on $\pi$. 


