On torsion units in integral group rings of Frobenius groups

For a finite group $G$, let $\tilde{\mathbb{Z}}$ be the semilocalization of $\mathbb{Z}$ at the prime divisors of $\mathrm{\mid }G\mathrm{\mid }$. If $G$ is a Frobenius group with Frobenius kernel $K$, it is shown that each torsion unit in the group ring $\tilde{\mathbb{Z}}G$ which maps to the identity under the natural ring homomorphism $\tilde{\mathbb{Z}}G\to \tilde{\mathbb{Z}}G\mathrm{/}K$ is conjugate to an element of $G$ by a unit in $\tilde{\mathbb{Z}}G$.