  The contact process is a simple model for the spread of an infection in a
structured population. We consider a variant of this process on
Bienaym\'e-Galton-Watson trees, where vertices are equipped with a random
fitness representing inhomogeneous transmission rates among individuals. In
this paper, we establish conditions under which this inhomogeneous contact
process exhibits a phase transition. We first prove that if certain mixed
moments of the joint offspring and fitness distribution are finite, then the
survival threshold is strictly positive. Further, we show that, if slightly
different mixed moments are infinite, then this implies that there is no phase
transition and the process survives with positive probability for any choice of
the infection parameter. A similar dichotomy is known for the contact process
on a Bienaym\'e-Galton-Watson tree. However, we show that the introduction of
fitness means that we have to take into account the combined effect of fitness
and offspring distribution to decide which scenario occurs.
