I'd disagree with that and say that economically the correct response would be to put all the increase on to fuel.
This would further restrict demand and lower the amount oil exporting countries could get for the oil, so effectively keeping the excess money in the oil consuming nations.
I have my doubts that that will be politically possible though, and even more doubt that in a recession when payments for unemployment benefit etc rise that the result will be other than large increases in direct taxation.

You're right, that does make more sense. As you point out though, it could be politically untenable. Governments can only force down the bitter medicine if their electorate recognize the disease; this requires a well-educated workforce.