  We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of Markov
Fourier multipliers acting on a group von Neumann algebra $\mathrm{VN}(G)$
associated to a locally compact group $G$ can be dilated by a weak* continuous
group of Markov $*$-automorphisms on a bigger von Neumann algebra. Our
construction relies on probabilistic tools and is even new for the group
$\mathbb{R}^n$. Our results imply the boundedness of the McIntosh's
$\mathrm{H}^\infty$ functional calculus of the generators of these semigroups
on the associated noncommutative $\mathrm{L}^p$-spaces.
