It was also shown pictures of Thomas wearing a white KKK hood, while posing in front of the confederate flag holding a machete.
Both were found guilty Monday of being members of National Action, the extreme right-wing organisation that virtually deifies Adolf Hitler and the Third Reich.
Thomas was also found guilty of possessing terrorist material which included bomb-making instructions.
The Crown's case was that after being banned by the government in December 2016, National Action simply "shed one skin for another" and "rebranded".
The court heard that Thomas, who gave their baby the middle name Adolf, posed for a photo with the child while wearing Ku Klux Klan robes.
Jurors heard evidence of social media chats involving Thomas, Patatas and Bogunovic, discussing what prosecutors have alleged was the banned group's continuing operation, under a different name.
"We now know they were a risky, well-structured organisation", Midland Police Detective Chief Superintendent Matt Ward said in the police statement.
Intercepted Skype messages showed that Thomas rejected attempts by leaders to disband the group.
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"These members continued to believe in its racist neo-Nazi worldview, remained in contact on encrypted messaging apps, and organised meetings to keep the group going".
The police announced that four others were convicted on similar charges, three of whom had confessed to their involvement with National Action before the trial.
Three other men have also been convicted of being part of National Action having been arrested at the same time as the previous trio on January 3.
Thomas's close friend Darren Fletcher, 28, of Kitchen Lane, Wednesfield, West Midlands, together with Joel Wilmore, 24, of Bramhall Road, Stockport, Greater Manchester, and Nathan Pryke, 26, of Dartford Road, March, Cambridgeshire, all admitted National Action membership before the trial.
Vehvilainen, who was born in Finland, had served with the Royal Anglian Regiment with distinction since 2012.
Prosecutors said Thomas had a "fanatical and tribal belief in white supremacy".
"We have seen many convictions over the past few years in connection with Syria-related terrorism and this work continues apace".
Vehvilainen - a 34-year-old lance corporal in the army - was jailed for eight years in April.