  Nematicity in the heavily hole-doped iron pnictide superconductors remains
controversial. Sizeable nematic fluctuations and even nematic orders far from a
magnetic instability were declared in RbFe$_2$As$_2$ and its sister compounds.
Here we report a systematic elastoresistance study of series of isovalent- and
electron-doped KFe$_2$As$_2$ crystals. We found divergent elastoresistance upon
cooling for all the crystals along their [110] direction. The amplitude of
elastoresistivity diverges if K is substituted with larger ions or if the
system is driven towards a Lifshitz transition. However, we conclude none of
them necessarily indicates an independent nematic critical point. Instead, the
increased nematicity can be associated with another electronic criticality. In
particular, we propose a mechanism how elastoresistivity is enhanced at a
Lifshitz transition.
