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?

2$\begingroup$ I tried to input a diagram, but they seem to do not work here. Is it possible to use diagrams in MO? $\endgroup$ – Sergei Akbarov Sep 15 '14 at 18:00
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 nonnormal subgroup. The restriction of the permutation rep of $G/H$ to $H$ is nontrivial, so it can't factor through the trivial rep.

$\begingroup$ Ben, can this be true when $H$ is normal? $\endgroup$ – Sergei Akbarov Sep 18 '14 at 16:44

$\begingroup$ When $H$ is normal, then $\rho\circ \iota$ is a bunch of copies of $\pi$, so there the desired factoring is true. $\endgroup$ – Ben Webster♦ Sep 19 '14 at 2:34

$\begingroup$ My $H$ is normal. Ben, I need a reference. I must refer to a textbook or to a paper in my text. $\endgroup$ – Sergei Akbarov Sep 19 '14 at 13:21
In addition to @BenWebster's points, my own reaction would be to see that the hopedfor 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 hopedfor, which, especially in light of counterexamples, is hard to see the reason to believein.
Thus, I'd ask if there's a context in which something of this sort arises, so that perhaps what might really suffice/work/solvetheproblem is different enough so as to avoid the "unnaturality"/typeviolation in the first place, and avoid the tangible counterexample schema mentioned by Ben W... ?