Let p $\neq 2$ be a prime number. Let $G=\mathbb{H}(\mathbb{Z}_p)$ be the group of uni-triangular 3x3 matrices wih entries in the ring of p-adic integers, sometimes called the profinite three dimensional Heisenberg group. Consider the sequence of compact open subgroups given by $G_n := \mathbb{H}(p^n\mathbb{Z}_p)$, which is a basis of neighbourhoods at the identity of $G$. It is well known that any unitary irreducible representation $\pi \in \widehat{G}$ of $G$ has a non trivial kernel containing one of the subgroups $G_n$, so that $\pi$ also defines a representation of the finite group $G/G_n \cong \mathbb{H}(\mathbb{F}_{p^n})$, where $\mathbb{F}_{p^n}$ denotes the finite field with $p^n$ elements, and the unitary dual of $\mathbb{H}(\mathbb{F}_{p^n})$, the Heisenberg group over $\mathbb{F}_{p^n}$, has been already computed explicitly, see for instance:
MISAGHIAN, M. The representations of the Heisenberg group over a finite field. Armen. J. Math. 3 (2011), 162–173.
So the question is, can we obtain explicitly the unitary dual of $\mathbb{H}(\mathbb{Z}_p)$ by using somehow the already known representations of the quotient groups $G/G_n \cong \mathbb{H}(\mathbb{F}_{p^n})$?. More generally, can we obtain the unitary irreduciblre representations of a profinite group $G$ from the representations of the quotient groups $G/H$, where $H$ is a compact open subgroup of $G$? What if we restrict ourselves to some class of profinite groups, nilpotent groups or pro-p groups?
