Let $G$ be a locally profinite group and let $H$ be a closed normal subgroup. Let $\pi\_1$ and $\pi\_2$ be two smooth complex representations of $G$. Is there always a spectral sequence as follows?
$$E\_2^{p,q} = \operatorname H^p(G/H,\operatorname{Ext}\_H^q(\pi\_1,\pi\_2)) \implies \operatorname{Ext}\_G^{p+q}(\pi\_1,\pi\_2)$$
Here $\operatorname H^{\bullet}(G/H,\cdot) \simeq \operatorname{Ext}\_{G/H}^{\bullet}(\mathbb C,\cdot)$ denotes the group cohomology in the context of smooth representations.


I did find a paper [On the Continuous (Co) Homology of Locally Profinite Groups and the Künneth Theorem](https://doi.org/10.1006/jabr.1994.1028) by Hitta proving the existence of a Hochschild–Serre spectral sequence in the case $\pi\_1 \simeq \mathbb C$ (Theorem 2 loc. cit.). But what about arbitrary smooth $\pi\_1$?



For locally profinite groups $H\lhd G$, is there a spectral sequence $\newcommand\@[2]{{\rm Ext}_#1^{#2}(\pi_1,\pi_2)}H^p(G/H,\@Hq)\implies\@G{p+q}$?
I found a way to prove the existence of such a spectral sequence, at least when $H$ is open in $G$. In order not to leave this question unanswered, let me write things down.


If $G$ is a locally profinite group, let $\mathcal R(G)$ denote the category of smooth representations of $G$. If $V$ is an abstract representation of $G$, I will write $V^{G-\mathrm{sm}}$ for the subspace of smooth vectors. Thus $V^{G-\mathrm{sm}}\in \mathcal R(G)$.


Let $(\pi\_1,V) \in \mathcal R(G)$. The functor $\mathcal R(G) \to \mathcal R(G)$ mapping $(\pi\_2,W)$ to $\mathrm{Hom}(\pi\_1,\pi\_2)^{G-\mathrm{sm}}$ is exact and preserves injectives, see for instance the theorem in the section III.1.15 of the book by Renard and Schwartz. Here, the $\mathrm{Hom}$ space consists of $\mathbb C$-linear morphisms $V\rightarrow W$, and the $G$-action is given by
$$gf: v \mapsto g\cdot f(g^{-1}\cdot v).$$
If we compose this functor with taking the $G$-invariants, we obtain the functor $\mathcal R(G) \to \mathrm{Ab}$ given by $(\pi\_2,W) \mapsto \mathrm{Hom}\_G(\pi\_1,\pi\_2)$. Thus, we deduce that
$$\mathrm{Ext}^i\_G(\pi\_1,\pi\_2) = \mathrm H^i(G,\mathrm{Hom}(\pi\_1,\pi\_2)^{G-\mathrm{sm}}).$$
Now, by the paper of Hitta which I refer to in my question, we have a Hochschild-Serre spectral sequence computing the right-hand side.
$$E\_2^{i,j} = \mathrm H^i(G/H,\mathrm H^j(H,\mathrm{Hom}(\pi\_1,\pi\_2)^{G-\mathrm{sm}})) \implies \mathrm H^i(G,\mathrm{Hom}(\pi\_1,\pi\_2)^{G-\mathrm{sm}})$$


Inside the $E\_2^{i,j}$ term, one must be a litte bit careful. Namely, we want to replace $\mathrm H^j(H,\mathrm{Hom}(\pi\_1,\pi\_2)^{G-\mathrm{sm}})$, seen as a representation of $H$ by restriction, with $\mathrm{Ext}\_H^j(\pi\_{1|H},\pi\_{2|H}) \simeq \mathrm H^j(H,\mathrm{Hom}(\pi\_{1|H},\pi\_{2|H})^{H-\mathrm{sm}})$.


But $\mathrm{Hom}(\pi\_1,\pi\_2)^{G-\mathrm{sm}}\_{|H}$ consists of all linear morphisms $f:V\to W$ such that there exists $K \subset G$ an open compact subgroup with $f(k\cdot) = k\cdot f(\cdot)$ for all $k \in K$, and equipped with the action of $H$ restricted from $G$.  

And $\mathrm{Hom}(\pi\_{1|H},\pi\_{2|H})^{H-\mathrm{sm}}$ consists of all linear morphisms $f:V\to W$ such that there exists $K' \subset H$ an open compact subgroup with $f(k'\cdot) = k'\cdot f(\cdot)$ for all $k' \in K'$, and equipped with the natural action of $H$.


Clearly the former is a subspace of the latter. If we assume that $H$ is open, then any $K'$ is also open compact in $G$, so that the two spaces are equal. Then, the Hochschild-Serre spectral sequence above is the desired result.


