The Black Caps ended the second day well-placed at 303 for two, just 86 runs behind England's first innings 389.
Williamson and Ross Taylor (47 not out) had so far shared an unbroken stand of 155 after coming together when the tourists had lost two wickets with their score on 148.
But Williamson was given a reprieve late in the day when he went down the pitch to off-spinner Moeen Ali on 92 only for Jos Buttler, still more of a batsman than a wicketkeeper, to miss the stumping chance, albeit the ball spun sharply.
By contrast, with the new ball available after three overs' play on Saturday, England will aim for an early breakthrough on Saturday.
"It's New Zealand's day, definitely. They batted really well and we found it hard to take wickets," Ali told Sky Sports.
"Jimmy (Anderson) bowled quite well this morning, and (Stuart Broad), but maybe a little short on that pitch.
"We're not too far from the new ball tomorrow so hopefully we can make some inroads," added Ali, who ended Friday with figures of one for 52 in 17 overs.
Earlier, Martin Guptill made 70 in his first Test in two years and was well-supported by Tom Latham (59) during what was just New Zealand's fifth century opening partnership in a Test in England.
On a sunny day, and without much movement, England found life tough, with captain Alastair Cook not often straying from orthodox field settings.
The 24-year-old Williamson drove down the ground impressively and also used the cut shot to good effect in the first of this two-match series.
His fifty came in good time, off 78 balls with six fours.
But they lost their final three wickets for just 37 runs in 10.5 overs.
Left-arm paceman Trent Boult (four for 79) and debutant fast bowler Matt Henry (four for 93) leading the way.
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