During which years was the fleur de lys part of the English flag?

The English flag (St George's Cross) was adopted in 1191 I can find no reference that says it ever contained a fleur de lys.
As England was invaded by the Normans in 1066 it might be fair to assume the Normans brought it with them, but they didn't as their flag/arms contained lions. The fleur de lys, whilst an ancient symbol, wasn't used by French royalty on their flags until the 12th century.
There is a use of the fleur de lys in England and that is on a Royal Coat of Arms, Edward III quartered the English Arms with the French to show his claim over France in 1340.
The fleur de lys was removed from the Coat of Arms in 1801 when George III abandoned his claim to the French throne.

https://en.wikipedia.org/wiki/Flag_of_England
https://en.wikipedia.org/wiki/Fleur-de-lis#Frankish_to_the_French_monarchy
https://en.wikipedia.org/wiki/Royal_coat_of_arms_of_the_United_Kingdom#Kingdoms_of_England_and_Scotland

The fleur de lys is used on a couple of county flags. Leicestershire has one and Monmouthshire has three, although that is in Wales.

https://abcounties.com/flags/tag/fleur-de-lys/

The flags are modern, although they are based on the coats of arms of the counties. As pointed out, the use of the symbol was intended to show that France was an English possession. Apparently, it also represented the sixth son.

https://www.fleurdelis.com/symbolism.htm