These are the most important settings, but it's possible you'll need to fix more, too.
(You can compare the rest of the settings to most other hairs - but make sure to copy settings from an hair_alpha TXMT and not from the plain "hair" TXMT, as this one is meant to get the rest of the texture from the skin texture, and isn't alpha enabled. If the cMaterialDefinition tab says "SimSkin" it's the wrong one).
You can export the settings from one TXMT to another (the export/import buttons in the TXMT), but make sure to copy/paste the "stdMatBaseTextureName" and possibly also the normal map texture if there is one (rarely hairs, but sometimes for clothes) from the original TXMT so it points to the correct texture. Mostly useful if the settings are very different from one file to the other.
The new group used the same TXMT as the other alpha group, but I managed to figure out that I'd copied the extra property set lines off 'hair' instead of 'hair_alpha' and so it was pointing to the wrong TXMT. Fixed now.