I believe that it is true that if we have a group $G$, and two copies $H_1$, $H_2$ of some group $H$ as subgroups of $G$, we can fix a representation $V$ of $H$ and have the situation:
$\operatorname{Ind}_{H_{1}} V \ncong \operatorname{Ind}_{H_{2}} V$
I believe my example is Klein-4 in $S_4$, you can take a normal copy generated by products of disjoint $2$-cycles $(12)(34)$, etc., and a copy $\{ (12)(34),(12),(34), 1 \}$ (or do $\mathbb{Z}_3 \times \mathbb{Z}_3$ in $S_9$ )
Anyway, if $H_1$ and $H_2$ are conjugate subgroups, then the induced representations will be isomorphic as can be seen by the character formula
$ 1/|H| \sum_{x} V(x^{-1}gx) $ = character of $\operatorname{Ind}_{H_{1}} V $ = character of $\operatorname{Ind}_{H_{2}} V$
Is this correct? Thank you