Let $G=GL_2(\mathbb{Q}_p)$, and let $K$ be a compact-modulo-center subgroup of $G$, $\rho$ an irreducible smooth representation of $K$.

**Question 1:** Is $\mathrm{ind}_K^G \rho$ cuspidal?

Here cuspidal is meant in the sense that matrix coefficients are compactly supported.

More precisely, I know already from Bushnell-Henniart, The Local Langlands Conjecture for $GL(2)$, Thm. 11.4, that $\mathrm{ind}_K^G \rho$ is irreducible cuspidal if the following condition holds: $g\in G$ intertwines $\rho$ if and only if $g\in K$. This condition is obviously necessary for irreducibility, but is it also necessary for cuspidality?

**Question 2:** For which groups $G$ other than $GL(2)$ does this result hold?