Jelly, 23 bytes

3R×L+“©®€‘ɓŒcn/€§ċ1;`Żạ

A monadic Link. Input is a list of corners of the cube as Cartesian co-ordinates (cube aligned with the co-ordinate system). Output is a list of integers, [faces, corners, edges].

Try it online!

How?

3R×L+“©®€‘ɓŒcn/€§ċ1;`Żạ - Link: list of lists, C          e.g. [[0,1,1],[1,1,0],[1,1,1],[0,0,0]] -- this could represent "FHGA"
3R                      - range of 3                           [1,2,3]
   L                    - length of C                          4
  ×                     - multiply                             [4,8,12]
     “©®€‘              - list of code-page indices            [6,8,12]
    +                   - add                                  [10,16,24]
          ɓ             - start a new dyadic chain, f(C,X) where X is the above result
           Œc           - pairs of C                           [[[0,1,1],[1,1,0]],[[0,1,1],[1,1,1]],[[0,1,1],[0,0,0]],[[1,1,0],[1,1,1]],[[1,1,0],[0,0,0]],[[1,1,1],[0,0,0]]]
              /€        - reduce €ach with:
             n          -   (vectorising) not equal?           [[1,0,1],[1,0,0],[0,1,1],[0,0,1],[1,1,0],[1,1,1]]
                §       - sum each                             [2,1,2,1,2,3]
                 ċ1     - count ones                           2
                   ;`   - concatenate with itself              [2,2]
                     Ż  - prepend a zero                       [0,2,2]
                      ạ - absolute difference with X           [10,14,22]

If the corners must be "ordered" as they are in the question then this works with integers 0-7 as A-H for 25 bytes: 3R×L+“©®€‘ɓŒc^/€ḟ2<5S;`Żạ (reduces using XOR, filters out twos, then counts those less than five).

Perl 6, 59 58 bytes

{6+@_,|((2,3 X*4+@_)X-(@_ X~@_)∩~<<ords "% 286
>C/")}

Try it online!

Uses the numbers 0 to 7 to represent the corners. I probably should have matched them up to the same order as in the question... oops? Outputs a list in the order faces, corners, edges.

JavaScript (Node.js), 84 bytes

a=>[a.map(u=>a.map(v=>j-=!!'ABCDAEFGHEFBCGHD'.match(u+v),n++,j+=2),n=j=6)|2+j,n+j,n]

Try it online!

Charcoal, 48 45 bytes

≔Ｅθ↨℅ι²η≔⊘№⭆η⭆ηΣＥι↔⁻§λξν1ηＩＥ⟦⁶⁻⁸η⁻¹²η⟧⁺ι×⊕κＬθ

Try it online! Link is to verbose version of code. Uses digits 0-7 to represent the letters ABDCEFHG in the diagram. Outputs in the order faces, corners, edges. Explanation:

≔Ｅθ↨℅ι²η

Take the ASCII code of each character and convert it to base 2.

≔⊘№⭆η⭆η

Take the cartesian product of the list of base 2 numbers with itself.

ΣＥι↔⁻§λξν1η

XOR the pairs of base 2 numbers together and sum the number of 1 bits. Count how many pairs have a sum of 1 and divide that by 2. This gives the number of coincident corners.

ＩＥ⟦⁶⁻⁸η⁻¹²η⟧⁺ι×⊕κＬθ

Calculate and print the number of faces, corners and edges.