Let $G$ be a finite group, and let $H_1$, $H_2$ be two non-conjugate subgroups. Can we have $\text{Ind}_{H_1}^G(\mathbf{1})\cong \text{Ind}_{H_2}^G(\mathbf{1})$ as representations?
Using characters, this can be rephrased as, "If for all $g\in G$, $g$ fixes an equal number of left $H_1$ cosets as left $H_2$ cosets, must $H_1$ be conjugate to $H_2$ as subgroups?"
It is easy to show that this holds if either $H_i$ is normal, or cyclic, but the general case doesn't seem clear.
If the answer to this is false, can we say anything about such a pair $H_1$, $H_2$? For instance, by considering the identity, we can see that $|H_1|=|H_2|$.
Yes. A triple $(G, H_1, H_2)$ with this property is called a Gassmann-Sunada triple; the Wikipedia article gives an example with $G = SL_3(\mathbb{F}_2)$. They can famously be used to construct examples of pairs of isospectral Riemannian manifolds which are not isometric, and pairs of number fields with the same Dedekind zeta function which are not isomorphic.