  The analogue of the Riesz-Dunford functional calculus has been introduced and
studied recently as well as the theory of semigroups and groups of linear
quaternionic operators. In this paper we suppose that $T$ is the infinitesimal
generator of a strongly continuous group of operators $(\mathcal{Z}_T(t))_{t
\in \mathbb{R}}$ and we show how we can define bounded operators $f(T)$, where
$f$ belongs to a class of functions which is larger than the class of slice
regular functions, using the quaternionic Laplace-Stieltjes transform. This
class will include functions that are slice regular on the $S$-spectrum of $T$
but not necessarily at infinity. Moreover, we establish the relation of $f(T)$
with the quaternionic functional calculus and we study the problem of finding
the inverse of $f(T)$.
