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Ebbsfleet United's players and officials were subjected to "abusive" and "aggressive" behaviour by some of their own supporters at Hemel Hempstead on Saturday.
The game finished 1-1, with Fleet conceding an 82nd-minute equaliser.
In a statement on the club's website, Ebbsfleet vice-chairman Peter Varney said: "The behaviour of a small minority of fans at Hemel Hempstead on Saturday was unacceptable.
"We fully respect the right of any supporter to criticise and to air their opinions good and bad. What we will not tolerate is officials and players being subjected to supporters approaching them in an aggressive way using foul and abusive language.
"I have said on many occasions that our supporters have a vital role to play in helping us achieve our ambitions but this small minority is creating a negative atmosphere at away matches.
"All of us are very approachable and happy to have any discussion with supporters at any time.
"It is particularly sad to see players feeling unable to acknowledge our own fans at the final whistle.
"I would urge those involved to stop this behaviour to avoid the need for any further action."