  We give a simple way to study the isotypes of the homology of simplicial
complexes with actions of finite groups, and use it for Milnor fibers of
\textsc{icis}. We study the homology of images of mappings $f_t$ that arise as
deformations of complex map germs $f:(\mathbb{C}^n,S)\to(\mathbb{C}^p,0)$, with
$n<p$, and the behaviour of singularities (instabilities) in this context. We
study two generalizations of the notion of image Milnor number $\mu_I$ given by
Mond and give a workable way of compute them, with Milnor numbers of
\textsc{icis}. We also study two unexpected traits when $p>n+1$: stable
perturbations with contractible image and homology of $\text{im} f_t$ in
unexpected dimensions. We show that Houston's conjecture, $\mu_I$ constant in a
family implies excellency in Gaffney's sense, is false, but we give a proof for
the cases where it holds. Finally, we prove a result on coalescence of
instabilities for the cases $(n,p)=(1,2), (2,3)$, showing that it is false in
general.
