Mackey theory in the setting of locally profinite groups

Let me reproduce and translate the relevant results from Vignéras's book (keeping the same notations). Putting these results together yields some generalisation of what you stated, but you have to be careful with the two types of induction. $\newcommand{\Mod}{\mathrm{Mod}} \newcommand{\Res}{\mathrm{Res}} \newcommand{\Ind}{\mathrm{Ind}} \newcommand{\ind}{\mathrm{ind}}$

Let $R$ be a commutative unital ring. Let $G$ be a locally profinite group. Let $\Mod_R(G)$ denote the category of smooth $RG$-modules.

Section 5.1:

Let $H$ be a closed subgroup of $G$. Let $$ \Res_{H,G}\colon \Mod_R(G) \to \Mod_R(H) $$ denote the restriction functor.

Let $$ \Ind_{G,H}, \ind_{G,H} \colon \Mod_R(H) \to \Mod_R(G) $$ respectively denote smooth induction and smooth compactly supported induction.

Section 5.4:

Let $H,K$ be closed subgroups of $G$. $$ I_{K,H} = \Ind_{K,K\cap H}\Res_{K\cap H,H} \text{ and }i_{K,H} = \ind_{K,K\cap H}\Res_{K\cap H,H}. $$

For $\sigma\in \Mod_R(H)$ and $g\in G$, let $g(\sigma)$ denote the representation of $g(H) = gHg^{-1}$ defined by $$ g(\sigma)(ghg^{-1}) = \sigma(g). $$

Section 5.5:

Let $H,K$ be closed subgroups such that all double cosets $HgK$, $g\in G$ are open and closed.

For all $\sigma\in \Mod_R(H)$ we have isomorphisms $$ \Res_{K,G}\Ind_{G,H}(\sigma) \cong \prod_{HgK}I_{K,g(H)}g(\sigma) $$ and $$ \Res_{K,G}\ind_{G,H}(\sigma) \cong \bigoplus_{HgK}i_{K,g(H)}g(\sigma). $$

If $H$ or $K$ is open, then the double cosets are open and closed.

Section 5.6:

Let $H$ be a closed subgroup of $G$.

1. Let $W\in \Mod_R(H)$ be admissible. Then $\Ind_{G,H}W$ is admissible iff $\ind_{G,H}W$ is admissible, and if they are admissible then $\Ind_{G,H}W = \ind_{G,H}W$.
2. If $G/H$ is compact, then $\Ind_{G,H}=\ind_{G,H}$ preserves admissibility.

Section 5.7:

Let $H$ be a closed subgroup of $G$.

1. $\Ind_{G,H}$ is right adjoint to $\Res_{H,G}$.
2. If $H$ is open, then $\ind_{G,H}$ is left adjoint to $\Res_{H,G}$.

The answer to your question is the main theorem of :

Kutzko, P. C. Mackey's theorem for nonunitary representations. Proc. Amer. Math. Soc. 64 (1977), no. 1, 173–175.

You've got to be careful with the induction functors. In the setting of locally profinite groups, the "good" analogue of the induction functor of finite group representations is the compact-induction functor from an open compact subgroup.