Could you guys do this with added FR composite position (not just the 8 - U layer edges), and going out to 10-cycles? How about identifying those V.I.P. even#-cycle algs (2,4,6,8,10-cycle) that also keep all of the composite edges together?

Sorry if it's seemed like a lot of time to do these calculations. With the addition of a couple more edges, the number of permutations increases greatly. So while the calculations for 8 edges took only very few minutes, for 10 edges the calculation time increased to several hours.

I note that with the FR edges added, conjugations by U layer turns is used instead of conjugations by y-axis rotations.

So far, I've finished only the calculations for reducing cases by rotational symmetry. I'll plan to append additional results to the end of this post as I complete them.

For 10 edges, the number of cases for n-cycles for n={2,3,4,...,10} that involve at least 1 of the FR edges are:

{ 5, 32, 210, 1176, 5460, 20160, 55440, 100800, 90720 }

Adding in the previous results for 8 edges, the total numbers are:

{ 13, 60, 320, 1512, 6308, 21600, 56716, 100800, 90720 }

The above numbers include the cases of n-cycles that don't split up any pairs. The number of such cases that include permuting of the FR edges are:

{ 1, 0, 2, 0, 12, 0, 48, 0, 96 }

The total number of these n-cycle cases (whether or not FR edges are permuted) are:

{ 2, 0, 5, 0, 20, 0, 60, 0, 96 }

EDIT 1:

I note that I counted the number of 2-cycle cases that don't split up any edge pairs as 2. Those cases are (1) swapping the element in a U-layer edge pair, and (2) swapping the elements of the FR edge pair. Of course, these two cases could be viewed as the same thing. Generally, I could use cube rotations xy and y'x' to reduce cases that only involve the FR edge pair and 2 adjacent edge pairs in the U layer (assuming we're also allowing conjugation by U-layer moves; otherwise specifically the UF and UR edge pairs ). These symmetries have not been considered in my calculations. So the swapping of the two edges of a pair is counted as 2 cases instead of 1.

Results for counting reflections as the same case.

n-cycles involving at least one of the FR edges (for n=2,3,4,...,10):

{ 3, 16, 105, 588, 2730, 10080, 27720, 50400, 45360 }

total cases (for n=2,3,4,...,10):

{ 9, 30, 166, 756, 3170, 10800, 28382, 50400, 45360 }

n-cycles preserving edge pairing involving at least the FR edge pair:

{ 1, 0, 1, 0, 6, 0, 24, 0, 48 }

n-cycles preserving edge pairing (whether or not FR edges move):

{ 2, 0, 3, 0, 10, 0, 30, 0, 48 }

EDIT 2:

Results counting reflections and inverses as the same case.

n-cycles involving at least one of the FR edges (for n=2,3,4,...,10):

{ 3, 13, 55, 324, 1380, 5160, 13920, 25440, 22800 }

total cases (for n=2,3,4,...,10):

{ 9, 20, 93, 408, 1630, 5520, 14311, 25440, 22800 }

n-cycles preserving edge pairing involving at least the FR edge pair:

{ 1, 0, 1, 0, 5, 0, 14, 0, 30 }

**n-cycles preserving edge pairing (whether or not FR edges move):**

{ 2, 0, 3, 0, 8, 0, 20, 0, 30 }