  In this work, we consider self-similar profiles for Smoluchowski's
coagulation equation for kernels which are possibly unbounded perturbations of
the constant one. For this model, we show that the self-similar solutions for
the perturbed kernel are close in weighted $L^1$ spaces to the profile of the
unperturbed equation, i.e. the profiles are stable with respect to the
perturbation. Additionally, we revisit the problem of uniqueness for these
coagulation kernels. In fact, we will improve a corresponding result (see
arXiv:1510.03361 and ref. [22]) by relaxing the conditions on the perturbation
significantly while at the same time the corresponding proof can also be
notably shortened.
