  We prove that for a free product $G$ with free factor system $\mathcal{G}$,
any automorphism $\phi$ preserving $\mathcal{G}$, atoroidal (in a sense
relative to $\mathcal{G}$) and none of whose power send two different
conjugates of subgroups in $\mathcal{G}$ on conjugates of themselves by the
same element, gives rise to a semidirect product $G\rtimes_\phi \mathbb{Z}$
that is relatively hyperbolic with respect to suspensions of groups in
$\mathcal{G}$. We recover a theorem of Gautero-Lustig and Ghosh that, if $G$ is
a free group, $\phi$ an automorphism of $G$, and $\mathcal{G}$ is its family of
polynomially growing subgroups, then the semidirect product by $\phi$ is
relatively hyperbolic with respect to the suspensions of these subgroups. We
apply the first result to the conjugacy problem for certain automorphisms
(atoroidal and toral) of free products of abelian groups.
