Let $G$ be a profinite group, $H$ a normal subgroup of $G$ of finite index and $\rho:H\to {\rm GL}_n(\mathbb{Q}_p)$ a $p$-adic representation of $H$ with torsion-free image.
Question: Is the image of the induced representation $\text{Ind}_{H}^{G}(\rho):G\to {\rm GL}_{n\cdot [G:H]}(\mathbb{Q}_p)$ also torsion-free?
The answer is no.
We have an almost trivial counter-example: let $\rho$ be the trivial representation of $H$, then of course the image of $\rho$ is torsion-free! However, the induced representation of the trivial representation to $G$ is the permutation representation on the set of cosets $G/H$, whose image is a non-trivial finite group and obviously contains non-trivial torsion.
But we have the following: the image of the induced representation $\mathrm{Ind}_{H}^{G}(\rho)$ contains no torsion element with order prime to $m=[G:H]$.