  We describe $\frac{1}{2}$-derivations, and hence transposed Poisson algebra
structures, on Witt type Lie algebras $V(f)$, where $f:\Gamma\to\mathbb C$ is
non-trivial and $f(0)=0$. More precisely, if $|f(\Gamma)|\ge 4$, then all the
transposed Poisson algebra structures on $V(f)$ are mutations of the group
algebra structure $(V(f),\cdot)$ on $V(f)$. If $|f(\Gamma)|=3$, then we obtain
the direct sum of $3$ subspaces of $V(f)$, corresponding to cosets of
$\Gamma_0$ in $\Gamma$, with multiplications being different mutations of
$\cdot$. The case $|f(\Gamma)|=2$ is more complicated, but also deals with
certain mutations of $\cdot$. As a consequence, new Lie algebras that admit
non-trivial ${\rm Hom}$-Lie algebra structures are found.
