Police are said to be investigating after members of the group uploaded pictures to Twitter of them holding the National Action flag while performing the gesture at the complex near Weimar last week.
The picture, captioned 'Oy vey, such horrors. Dem bois recently on tour in Germania', has been retweeted 14 times and has gained 46 Likes.
Two members of the group gave the banned salute in a subterranean ante room of the former crematorium known as the "corpse cellar" where bodies of the dead were stored before incineration.
Public displays of Nazi salutes are strictly forbidden under Germany's postwar constitution and the National Action members could be jailed for up to six months if they are identified.
The members pixilated their faces in the picture uploaded to the National Action NorthWest twitter page.
"Meat Hooks", and "execution room Buchenwald 2016" was scrawled on the photo, something that did not go unnoticed by Germany's Bild newspaper.
It wrote: "What was especially perfidious was the fact that next to large hooks from which the SS hung the bodies the neo-Nazis wrote 'meat hook' next to it."
The custodians of Buchenwald, which is visited by half a million tourists and Holocaust survivors every year, are said to have filed a complaint to police accusing the National Action members of disturbing the peace of the dead.
Dr. Volkhard Knigge, director of the memorial site, has been quoted as saying: "The British bro-nazis displayed a specific criminal energy in the corpse cellar of the camp.
"They openly identified with the Nazis and their mass murder programme. This is a grave insult to the 280,000 people who were imprisoned here, some of them British."
He added that he believed the act was not spontaneous but staged to cause the maximum amount of outrage possible.
According to The Sun, a police spokesman confirmed police were investigating the matter.
"We take this very seriously and are investigating to identify the culprits," the spokesman was quoted as saying.
National Action are infamous for their failed attempts to stage the 'White Man March' in Liverpool.
In August last year members of the group, having earlier threatened Liverpool would "go up in flames" if their march was stopped, had to take shelter in a lost property depot at a train station after becoming overwhelmed by counter protesters who pelted them with rubbish.
The group were further ridiculed earlier this month when their deputy leader, Alex Davies, was laughed out of Bath after admitting he didn’t know what a white person looked like.