If our involution is 1032456, the shifted versions of this involution are all present, including 0124356. Apply this, then the original involution, to get two transpositions on either end and a 3 cycle in the middle. Square this to get a 3 cycle, which implies A7.
Let 01 swap as above, along with 34. A shifted version swaps 23 and 56. Together they produce a 3 cycle, as above. If we're looking for something other than A7, at least one of the transpositions does not swap adjacent elements.
Swap 01 as above, and let 2 swap with something other than 3. If it swaps with 4 or 5 we generate the simple group of order 168. If 2 and 6 swap we are back to two adjacent transpositions.
The last involution swapping 01 swaps 35. This can be combined with a shifted version of itself to give a 3 cycle. Now we know none of the transpositions swap adjacent elements.
If a transposition has a gap of 1, shift it around to swap 02. The other transposition may also have a gap of 1, say 35. This combines with a shifted version of itself to give a 3 cycle. If we swap 13, a shifted version gives 0125634. Combine this with the original to build another 3 cycle. Thus at least one of the transpositions has a gap > 1.
Swap 02 as above, and 14. This was pattern z, when generating the group of order 168. The last involution with 02 swaps 36. This is more commonly presented as 4321056. Shift two to the right, 0165432, and apply to the original to build a 3 cycle. That takes care of the gap of 1.
Both transpositions have a gap of 2, or 3, depending on what you call the inside. Let one of them swap 03. The other acts on 1 or 2. Reflect as necessary, so that it acts on 1. It either swaps 14 or 15. Both possibilities allow us to build a 3 cycle.
The group spanned by the 7 cycle and the double transposition is either A7 or G168.