ClearAll[Primediffs]
p = Prime[Range[PrimePi[10^6]]];
Primediffs[seq_] := {First[#], Last[#], Length[#]} &[p[[#1 ;; #2 + 1]] & @@@ SequencePosition[Differences[p], seq]]
Primediffs[{2}]
Primediffs[{1}]
Primediffs[{2, 2}]
Primediffs[{2, 4}]
Primediffs[{4, 2}]
Primediffs[{6, 4, 2}]