Notation
Faces: The usual U D L R F B
Outer Layers: U D L R F B
Inner Layers: u d l r f b
Corners: Defined by the three converging faces in any order eg FUR=RUF etc
Centres: Defined by Face then Quadrant of face (so A is FUR, B is RUF and C is UFR)
Pips: Defined by Face, Inner Layer, Outer Layer (so pip above A is FrU, pip to right of A is FuR)
Edges: Can be specified by either pip
Moves: Are considered clockwise when viewed from the face nearest the move

Centres
Centres A B and C centres are linked and form the UFR corner of a 2x2x2.

Solve centres as 2x2x2 using slice moves only. (In real puzzle you will probably find it easier to twist half the puzzle at once)

Pips
These are the circular half-segments.
Pips Pips are connected in fixed pairs (the sharp corners point towards each other)
eg
  • UrF:RuF
  • FrU:RfU
 
U and D pips
Solve U and D pips using this modified 3x3x3 sequence and its reflection
Sequence A:r U r' U r U2 r'  - puts the RfU:FrU pair into the UrF:RuF position
or its reflection
Sequence B:l' U' l U' l' U2 l  - puts the LfU:FlU pair into the UlF:LuF position
Pip moveThe effect of  Sequence A.
Start with red pip on the right face.		 Sequence BThe effect of  Sequence B.
Start with red pip on the left face.

1) Find a top face pip (red in this example) in the top layer, position it correctly by turning the upper layer and apply the relevant sequence. Repeat if possible with other pips.

2) When all those pips are done find top face pips in other layers of faces L R F B. Get them into the top layer with a inner layer move, followed by U move(s) to move the pip out of that layer, then complete with the reverse inner layer move. Use step 1 to solve the pip and repeat for other pips.
Step 2l2 U' l2 Example of step 2


3) Finally if any top face pips are in the bottom face, invert the cube (top to bottom) and move them from the new top layer using sequence A or B above. Once all pips are in faces L R F B reinvert the cube and solve as in step 2.

4) Check all pips in u layer are correct. If not remove them from the top layer using sequence A or B, then solve them using step 1.
Top face pip in wromg position Top face pip UrF is in wrong position (notice green at RuF)
Sequence A places the pair of pips ready for step 1


Now repeat steps 1, 2 and 4 for the opposite face (orange in this case). Step 3 shouldn't apply.

Solve remaining pips as 3x3x2 edges
8 pairs of pips left
Front and Back layers may be moved freely. Restrict other moves to l2 r2 u2 d2		 3x3x2 reference


Edge Pairing
Perform edge pairing as for standard 4x4x4 (pips can easily be realigned after edge pairing is complete)

3x3x3 adapted sequences
The puzzles in now reduced to 3x3x3 (almost)
Getting the cross is slightly trickier than usual (in order to keep pips aligned)
Then remainder of first two layers is as normal
I then use beginner's method for last layer (keeps pips correctly aligned)

The only parity case with this method is swap UF-UB edge pairs (without swapping any pips)
r' F U' R F' U r
l U' F R' U F' l'