Depends what the FA view as acceptable.
We are still strong favourites to win the group, we beat the Czechs 5-0 last time we played them at home and they haven't looked very good in either game. So if we get Portugal in the R16 and lose (without getting battered and/or completely played off the park) would they shrug and say "ah, we got a tough draw in a stupid format"?
Or if we draw with/lose to the Czechs and come second, win against whoever we get in the R16 and then lose to France in the QF (same caveats as vs Portugal), would they reason that we don't generally get past that stage and they are the world champions and all that, harsh to give someone the bullet for that.
Then we'll obviously qualify for the World Cup with no difficulty at all as our qualification record is ridiculous and we're already top with none of the other teams likely to trouble us, Poland missed their chance at Wembley, Andorra AND San Marino are in our group FFS.
I don't see him being sacked any time soon is the long and the short of it, we're not a nation who has been grooming a replacement with the U21s for years - at least I hope not as it was Aidy fucking Boothroyd -, we're probably never going to go down the foreign route again due to how it ended each of the two times we did it and there are few viable English candidates available, it was only going to be a catastrophe here that would lead to that, and it's not especially likely to pan out that way, not now we're 99% assured of going through. Earliest you can hope for is in 18 months time. If you want that you also have to want us to have a shit World Cup, or that by then having been in the job for six and a bit years, he's had enough.