When is the induced representation factored through the initial one? Let $H$ be an open subgroup in a locally compact group $G$, $\iota:H\to G$ the embedding of $H$ into $G$, $\pi:H\to B(X)$ a unitary representation of $H$ in a Hilbert space $X$, and $\rho:G\to B(Y)$ the corresponding induced representation of $G$. When does there exist a (continuous in a proper sense) involutive homomorphism of algebras $\varphi:B(X)\to B(Y)$ such that 
$$
\varphi\circ\pi=\rho\circ\iota \quad ?
$$
This seems to be true at least when $H$ is finite and $\pi$ is irreducible. Is it possible that this is always true?
 A: This is false in many cases where $G$ is finite.  Let $\rho\circ \iota$ and $\pi$ denote the corresponding maps of group algebras $\mathbb{C}[G]$.  The equation above can only hold if any element killed by $\pi$ must also be killed by $\rho\circ \iota$, that is $\mathrm{ker}(\pi)\subset \mathrm{ker}(\rho\circ \iota)$.  If $\pi$ is irreducible, then $\mathrm{ker}(\pi)$ is the sum of all the other matrix algebra summands in the group algebra, so  $\mathrm{ker}(\rho\circ \iota)$ can only contain $\mathrm{ker}(\pi)$ if $\rho\circ \iota$ is just a bunch of copies of the same irrep.  Of course, this could happen, but it doesn't have to; for example, let $\pi$ be trivial, and $H$ any non-normal subgroup.  The restriction of the permutation rep of $G/H$ to $H$ is non-trivial, so it can't factor through the trivial rep.
A: In addition to @BenWebster's points, my own reaction would be to see that the hoped-for property is fighting against any/all of several natural characterizations of "induction", e.g., as right (or left) adjoint to restriction... That is, some "type" violation is hoped-for, which, especially in light of counter-examples, is hard to see the reason to believe-in.
Thus, I'd ask if there's a context in which something of this sort arises, so that perhaps what might really suffice/work/solve-the-problem is different enough so as to avoid the "un-naturality"/type-violation in the first place, and avoid the tangible counter-example schema mentioned by Ben W... ?
