  Given a (not necessarily continuous) homomorphism between Banach algebras
$\T\colon\A\to\B$, an element $a\in\A$ will be said to be B-Fredholm
(respectively generalized B-Fredholm) relative to $\T$, if $\T(a)\in \B$ is
Drazin invertible (respectively Koliha-Drazin invertible). In this article, the
aforementioned elements will be characterized and their main properties will be
studied. In addition, perturbation properties will be also considered.
