Save isn't moving over at all from .4...
As far as I can tell you have to move your save games yourself. Just make sure to move or copy your saves
BEFORE you delete the old version because the save folder location is actually within the subfolder where you extracted the game:
...\Folder where you run the game from\The Halo Project_Data\Saves
The NEW save game location (v0.51) is:
C:\Users\<
YOUR USER NAME HERE>\AppData\LocalLow\Hush\The Halo Project\Saves
This is information that the dev
SHOULD have made clear within either the v0.51 Changelog or by adding the information somewhere in the OP so as to inform everyone and, hopefully, try to prevent players from accidentally deleting all of their save games!
If, however, you've already deleted the old version, including your save games, then feel free to blame the dev for not warning everyone!