  We provide a complete classification of groups that can be realized as
isometry groups of a translation surface $M$ with non-finitely generated
fundamental group and no planar ends. Furthermore, we demonstrate that if $S$
has no non-displaceable subsurfaces and its space of ends is self-similar, then
every countable subgroup of $\operatorname{GL}^+(2,\mathbb{R})$ can be realized
as the Veech group of a translation surface $M$ homeomorphic to $S$. The latter
result generalizes and improves upon the previous findings of
Przytycki-Valdez-Weitze-Schmith\"{u}sen and Maluendas-Valdez. To prove these
results, we adapt ideas from the work of Aougab-Patel-Vlamis, which focused on
hyperbolic surfaces, to translation surfaces.
