Twist points:

BETA solver  

The point pieces are leading. Orient them toward the plane that borders all 3 pieces that have a certain color.

Bottom layer: complete the point-diamonds.

Edge pieces: find the fitting piece and bring it into the face of its not bottom color, with the bottom color in the plane. Bring the plane to the front.

If it is in the bottom already, knock it out with below algorithm(s). Being at the front of the bottom.

If the piece is to the left: L R' L' R   If the piece is to the right: R' L R L'. Or for us humans: "down down up up" starting with the side of the piece.

Top diamonds: align the top piece with the side pieces to complete the top-diamonds.

Middle layer: there are a few configurations possible.

If one piece is solved and the other 2 are (oriented) wrong, put the correct piece to the back and apply algorithm: R' L R L' U L' U' L

If the 3 pieces need to be shifted clockwise: R U' R' U' R U' R'. Note the U' usage.

If the 3 pieces need to move anti-clockwise: R U R' U R U R'. Note the U usage.

If none of the above apply, move one correctly oriented piece to the correct position with one of the above shift moves, and reassess.