The Nottingham Forest right-back stood in the way of a stinging effort from Rashford in the early moments of the first-half in Qatar.
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It struck him flush on the head and forced a stoppage in play, but Williams was initially passed fit to play on by the Dragons' medical staff.
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After Williams continued for a lengthy period of time, the former Liverpool defender went down again and Wales' medical team took the situation out of his hands.
The 21-year-old was desperate to play on in what could be his last match at the 2022 World Cup, with Wales needing an unlikely set of circumstances to progress through the group stage.
Williams could be seen asking to stay on the pitch, but he was eventually replaced by Connor Roberts.
It mirrored a situation England experienced in their opening match against Iran, when Alireza Beiranvand suffered a sickening blow to the head and continued before eventually being taken off.
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As per FIFA rules, if there is any suspicion of a player having sustained a concussion, they must be removed from the field of play and not allowed to return if they are deemed to be concussed.
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