Green modules are not diodes, are DPSS (diode pumped solid state) assemblies, and have output beam characteristics differents from diodes ..... and aixiz lenses have a FL (focal lenght) of 3 or 4 mm, set specifically for focus the output divergence typical of a diode, not the one from a crystal assembly of a green module .....
If you take away the cap of a green pen and put the aixiz lens in front of the output lens, the beam just become focused at the FL of the aixiz lens, then diverge again ..... you get a focus point where the beam burn, but no straight beam .....
If you remove also he front lens of the module and place an aixiz lens there, the FL of the aixiz lens is too short, and it still focus the beam near it (also if a bit more distant), then diverge again .....
The only way that you can obtain something like a collimated beam with an aixiz lens, is placing it very near to the expander, in this way you can obtain a thinner green beam, but this have 3 problems ..... first, these lens holders are usually assembled with loctite glue, and is a real pain you-know-where

to take them away without destroy the module or ruin the crystal / expander ..... second, the output diameters don't match, so you need to build a custom threaded holder for the aixiz lens ..... and third, the focusing process is a real nightmare, with these distances ..... i've tried something similar with an o-like module, and using a short FL as focusing lens, the process was so critic that for defocus it, was enough almost just look the lens bad, not speaking about moving it .....