No you don't, this has been debated many a time, and VERMIN is still being debated, fox obviously works, but so to does ANY OTHER LAWFUL QUARRY, and there is nothing in that that mentions fox!
Even if you were right about this then is fox lawful quarry with a 22lr?? it all comes down to YOUR police force
The Fox has NO calibre designation whatsoever in law, of course it is Lawful quarry, that has nothing to do with your individual region.
Getting an ANY OTHER LAWFUL QUARRY condition certainly does though!
Nice i'm going to go and shoot some Pheasants with my .22lr